DOMINIC WYNES-DEVLIN 200759087 1 FORMAL REPORT – ZEEMAN EFFECT Abstract The aim of this experiment was to observe the ‘normal Zeeman Effect’ and to then quantify this by determining the experimental value for the Bohr Magneton. The value was to determined to be μ B = (6.1 +/- 0.9) ×10 −24 J T -1 . A Cd (Cadmium) lamp was placed inside the magnetic field of an electro-magnet and an interference pattern was observed from the apparatus (setup) producing main-order interference rings and ‘orbiting satellite’ rings. The applied voltage to the Electromagnet was varied and the direction of the magnetic field (longitudinal and transverse positions) was rotated with respect to the horizontal track to observe how the interference pattern changed. A polarizer and a quarter wave-plate were then introduced to determine the components of the main rings and satellite rings of the interference pattern. 1. INTRODUCTION 1.1 ZEEMAN EFFECT Pieter Zeeman discovered that spectral lines of an atom begin to split into further lines when in the presence of a magnetic field. This effect become commonly referred to as the Zeeman effect and in … Lorentz provided a theoretical framework for this effect. However when Zeeman observed these splittings, he found further splitting levels that neither Lorentz nor classical physics could explain. These further unexpected splittings were dubbed the ‘Anomalous Zeeman effect’. The anomalous Zeeman effect was later explained through the application of quantum mechanics that considered the effects of electron spin – S. This ‘Anomalous Zeeman Effect’ is now considered the more fundamental process in that the ‘Normal Zeeman effect’ explained by Lorentz is only the case when the electron spin – S = 0. 1.2 QUANTUM NUMBERS AND QUANTUM THEORY The total angular momentum of an electron can be expressed by: = + [1] Where J is the total angular momentum, L is the orbital angular momentum and S is the electron spin. Since we are only considering the ‘Normal Zeeman Effect’ we do not consider the electron spin and hence S=0 = Therefore, the total angular momentum is equal to the orbital angular momentum of the electron going ‘around’ the nucleus. The spectral line of interest for the Cd atom corresponds to the transition between the 1 D 2 state to the 1 P 1 state. In the presence of the magnetic field the 1 D 2 state splits into 5 components for this level, and the 1 P 1 level splits into 3 components. Figure 1 shows the total number of splitting lines and the possible transitions between the states.
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DOMINIC WYNES-DEVLIN 200759087
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FORMAL REPORT – ZEEMAN EFFECT
Abstract
The aim of this experiment was to observe the ‘normal Zeeman Effect’ and to then quantify this by
determining the experimental value for the Bohr Magneton. The value was to determined to be
µB = (6.1 +/- 0.9) ×10−24 J T-1. A Cd (Cadmium) lamp was placed inside the magnetic field of an electro-magnet
and an interference pattern was observed from the apparatus (setup) producing main-order interference
rings and ‘orbiting satellite’ rings.
The applied voltage to the Electromagnet was varied and the direction of the magnetic field (longitudinal
and transverse positions) was rotated with respect to the horizontal track to observe how the interference
pattern changed. A polarizer and a quarter wave-plate were then introduced to determine the components
of the main rings and satellite rings of the interference pattern.
1. INTRODUCTION
1.1 ZEEMAN EFFECT
Pieter Zeeman discovered that spectral lines of an atom begin to split into further lines when in the presence
of a magnetic field. This effect become commonly referred to as the Zeeman effect and in … Lorentz
provided a theoretical framework for this effect. However when Zeeman observed these splittings, he found
further splitting levels that neither Lorentz nor classical physics could explain. These further unexpected
splittings were dubbed the ‘Anomalous Zeeman effect’. The anomalous Zeeman effect was later explained
through the application of quantum mechanics that considered the effects of electron spin – S. This
‘Anomalous Zeeman Effect’ is now considered the more fundamental process in that the ‘Normal Zeeman
effect’ explained by Lorentz is only the case when the electron spin – S = 0.
1.2 QUANTUM NUMBERS AND QUANTUM THEORY
The total angular momentum of an electron can be expressed by:
𝐽 = 𝐿 + 𝑆 [1]
Where J is the total angular momentum, L is the orbital angular momentum and S is the electron spin.
Since we are only considering the ‘Normal Zeeman Effect’ we do not consider the electron spin and hence
S=0 𝐽 = 𝐿
Therefore, the total angular momentum is equal to the orbital angular momentum of the electron going
‘around’ the nucleus.
The spectral line of interest for the Cd atom corresponds to the transition between the 1D2 state to the 1P1
state. In the presence of the magnetic field the 1D2 state splits into 5 components for this level, and the 1P1
level splits into 3 components. Figure 1 shows the total number of splitting lines and the possible transitions
between the states.
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Figure 1 – Diagram showing: a) the splitting states and allowed transitions for the Cd atom, b) the polarisation directions for sigma
minus, pi and sigma plus components. [2][3]
The electrons can only transition between different quantised orbital angular momentum values – L (i.e.,
integer values 0, 1 , 2 , etc.) and as such the electron can’t transition between states of the same value for L.
The allowed transitions correspond to 3 different components that are dependent upon the difference in
magnetic quantum numbers – ∆𝑀𝐽.
For the sigma components ∆𝑀𝐽 = ±1 and for the pi components∆𝑀𝐽 = 0. These components correspond
to the different spectral lines given out by the atom. [4]
The difference in energy - Ediff - between the adjacent splitting lines is given by equation 1:
𝐸𝑑𝑖𝑓𝑓 = 𝜇𝐵 𝐵 (1)
Where B is the magnetic field strength applied and 𝜇𝐵 is the Bohr magenton constant given by: 𝜇𝐵 = ℏ𝑒
2𝑚𝑒.
The accepted value for the ‘Bohr Magneton’ is given as - 9.27400 9994(57) x 10-24 J T-1 [5]
In this experiment ∆𝐸 and B can be determined and used to calculate and experimental value for the Bohr
magneton. Equation 2 is used to calculate 𝐸𝑑𝑖𝑓𝑓:
𝐸𝑑𝑖𝑓𝑓 = −𝛼∆𝛼
𝑛2 (2)
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2.1 Setup of apparatus and procedure
The experiment consists of a Cadmium (Cd) Lamp (4.) is connected to the power supply (1.) and is placed
between the electro-magnet (3.) – where the electromagnet is connected to its power supply (2). The
electro-magnet can be positioned such that the magnetic field is perpendicular to the horizontal track
(transverse). It can then be rotated so that the magnetic field is parallel to this horizontal track (longitudinal)
as well. Both configurations can be studied and compared.
The condenser lens (5.) passes the light coming from the Cd Lamp out parallel to the horizontal track. The
light is then passed through the ‘Fabry-Peron Etalon’ (7.) producing an interference pattern consisting of
circular fringes or “rings”. The light then passes through to the imaging lens (8.), and enlarges the image of
the interference pattern is then passed through to the interference filter (9.) which then filters so that only
wavelengths in the red wavelengths range are visible through the eye-piece (9.). Furthermore, this is to
ensure that other spectral lines produced from electron transitions where the electron spin is non-zero are
not present in the interference pattern. Figure 2 below shows the following setup of the apparatus.
The power supply of the Electromagnet can be used to vary the voltage applied to the electromagnet to
observe how the varying voltage affects the observed interference pattern. A polariser and quarter-wave
plate (reducer)(6.) is later introduced between the condenser lens and the Etalon (Fabry-Perot) to determine
the polarizations of each component (pi, sigma+, sigma -) of the emitted light in relation to the observed
interference pattern. (See sections 2.31 and 2.32 for an understanding of circular and plane polarisations
and how the quarter-wave plate and polarizer work.)
In section 2.4 the (ocular) eyepiece and interference filter is replaced with a CCD camera that is connected to
a computer. The intensity distribution against the distance (in pixels) along the camera is recorded and is
then viewed on the computer (see section 2.4 for more details on how the CCD camera works). This can be
used to accurately determine the difference between the positions of the interference fringes for different
values of the applied magnetic field strength (obtained by varying the applied voltage to the electromagnet).
This is used to determine the Bohr Magneton (see section 5. for more details on this).
Figure 2 - Diagram of the experimental apparatus and setup used to observe the ‘Zeeman Effect’.
Longitudinal position for the Electro-magnet
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2.3 Polarization and Quarter-wave-plate (reducer)
2.31 Linear and circular polarised light and polariser
When light passes through a polariser Light is now considered linearly polarised. The direction of polarisation
can be a superposition of two other electric fields in two different directions but will still be linearly polarised.
Circular polarised light differs from linearly polarised light in that it consists of 2 electric field components in
different directions that have a phase difference of pi/2 between them. At each point the vector sum of
these will produce the direction of linear polarisation however since these two waves have a phase
difference between them, the vector sum will rotate in a circle as each electric field oscillates in time.
Figure 3 shows images of linearly polarised light and circular polarised light.
2.32 Quarter-wave-plate (reducer)
A quarter wave-plate is a bire-fringent material which has a thickness chosen such that it will introduce a
phase difference of a quarter wave-length between the two electric field components of a linearly polarised
light. This means that a linearly polarised light source can become circularly polarised when light passes
through it. A quarter wave plate can therefore be used to convert between linearly polarised light and
circularly polarised light. [7]
2.4 CCD Camera
Figure 4 below shows the intensity pattern produced by the CCD camera when the magnetic field is applied.
The smaller peaks off the main sharp peaks are characteristic of the Zeeman splitting. The position of these
smaller peaks and the main peaks can determine the distance between the satellite rings and main rings for
a given magnetic field strength. This device was extremely sensitive and hence difficult to position in such a
way that these smaller peaks were noticeably resolved and distinct from the main rings’ intensities.
Figure 3 - Diagram showing a) linearly polarised light and b) circular polarised light. [6]
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Figure 4 – Intensity plot against distance in pixels for the light with the applied magnetic field. [8]
3. Observations of the Zeeman Effect
3.1 – Observed effect of applied magnetic field
When the magnetic field is applied in the transverse case the interference pattern produces main rings and
satellite rings. Where as, in the case of the longitudinal position no main rings are present since the pi
components are not present when the magnetic field is parallel to horizontal track. We can deduce that
main rings are therefore the pi components and the sigma components are the satellite rings.
3.2 – Varying the voltage through electromagnet
Increasing the voltage (hence the current) of the electromagnet results in the distance between the satellite
rings increasing. This causes the higher satellite rings of the lower order to become closer to the lower
satellite ring of the next higher order satellite ring and as such make some of the rings appear more closely
packed where as others appear greater spread out.
Figure 5 – Interference rings produced for: a) no applied magnetic field, an applied magnetic field in b) the transverse
position, c) the longitudinal position.
Figure 6 – Interference rings produced when applied voltage has increase for a) the transverse position, b) the longitudinal position.
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3.3 – Introducing Polarizer and Quarter-Wave plate
3.31 – Adding Polariser
When the polariser is introduced into the transverse setup, both the main rings and satellite rings appear
when the polariser is set at 0 degrees to the vertical axis; where as, when the polariser is turned to 90
degrees only the satellite rings are visible. This is because the main ring is linearly polarised in the vertical
direction and so when the polariser is at 90 degrees the light cannot pass through it. This is because the pi
components of the main rings are linearly polarised, where as the satellite rings are the sigma components
and are circularly polarised hence they can still pass through the polariser. These observations are shown in
figure 7 below.
3.32 – Polariser and Quarter-wave-plate
When the quarter wave-plate is introduced in the longitudinal setup, circularly –polarised light becomes
linearly polarised. Since no pi components are present in the longitudinal position only the satellite rings can
be observed. When the polariser is at 0 degrees both satellite-rings can be observed. However when the
polariser is at 45 degrees only the outer satellite ring is observed, whereas when the polariser is changed to
-45 degrees (45 degrees anti-clockwise) only the inner satellite ring is observed. We can deduce from figure
1b) that the sigma plus components are anti-clockwise circularly polarised (ACCP) and the sigma minus are
clockwise circularly polarised (CCP) such that when the polariser is at 45 degrees then only the CCP light can
pass through and hence the outer satellite ring is visible and is the sigma plus component. The sigma
minus component must therefore be the inner satellite ring.
Figure 7 - Image of Interference rings for transverse position with a) no polariser, the polariser is at b) 0 degrees to the vertical axis,
c) at 90 degrees to the vertical axis.
Figure 8 - Image of Interference rings with polariser and quarter wave-plate for the polariser at a) 0 degrees to the vertical
axis, b) 45 degrees to vertical axis , c) -45 degrees to the vertical axis.
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4. Calibration of Electro-magnet
The Electro-magnet is calibrated to determine a relationship between the applied current supplied to the
Electromagnet and the magnetic field strength between the two electromagnets. The magnetic field
strength between the electromagnets is measured using a Gauss meter which has a sensitivity of ± 1mT
(0.001T). The magnetic field strength was measured for varying current values from 5.05 to 8.88 A with
intervals of approximately 0.5 A. This data (shown in Table 1 below) is then used to plot the Calibration curve
for the applied current – I against the Magnetic field strength- B; shown in figure 8 below.
Table 1: Data for the applied current – I, magnetic field strength – B of the Electromagnets for current values range from
5.05- 8.88 A.
Figure 8 – Calibration curve of the applied voltage against the magnetic field strength of the electro-magnet.