Zeeman Effect Essay Example
 Category:Physics
 Document type:Assignment
 Level:Masters
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 Words:2047
Zeeman Effect
Abstract
Zeeman Effect gives the study of the magnetic field on light. The study involves the effect of external field that is exhibited on light. Before the evolution of quantum physics, it proved difficult the phenomenon of the splitting spectral lines that split when subjected under the influence of magnetic field. Quantum physics dictates that, particles energy would presume specific data values. The assertion of Pauli Exclusion Principle, elucidate the effect that the electron’s energy would shift depending on the external source that acts on it. The electrons would make transitions to various states making the energy to be either emitted or absorbed. This ascertains the description from the quantum physics that, in case there are two electrons in a given state, they may possess same energy in as long as they have different quantum numbers. These quantum numbers that forms the subject of this experiment are the principal, orbital and the magnetic quantum numbers. The aim of this experiment is to study the Zeeman splitting of mercury atoms of wavelengths 546.1 nm (_{}) and compare the experimental results with the theoretical prediction.
Introduction
Zeeman Effect is a phenomenon exhibited when the energy levels of given atoms are split whenever an external magnetic field is placed in its vicinity. The interaction between the magnetic field B and the magnetic momentum _{} causes the relative orientation of their energies. The amount of energy is shifted in a relationship has an effect on the nuclear magnetic resonance (NMR). As noted by (Zettili, 2001), the splitting is dependent on the nuclear orientation and its static magnetic field.
The atoms are excited at levels that are above the ground state. Inherently, the quantum physics depicts that, the energies of particles presume certain specific values. In addition, the assertion of Pauli Exclusion Principle notes that, it is almost impossible for two electrons to presume the same state at a go. Electrons moving to a higher energy level would absorb energy, while those moving to lower energy level would lose energy. Zeeman’s effect would be experienced when a magnetic field is applied between the excited state and the ground state. This effect causes a slight emission of photons possessing different energies. The splitting of the magnitude would depend on the magnetic field.
Theoretical background
This phenomenon was discovered way back in 1896 by ZeemanLorentz hence its name. Efforts by faraday to proof the viability of whether there are changes accompanied in the spectrum of coloured flames when subjected to a magnetic deemed futile. A Bohr picture of an atom was used to elucidate the occurrence. An electron that is placed in a magnetic field would experience a Lorentz force which would tend to change the orbit of a given electron that exhibits its energy (Kenyon, 2011). The movement of the electron determines the change in the energy whether it is negative or positive depending on the motion of the electron. In cases where the field is along the plane of a given orbit, it follows that the Lorentz force exhibited would be zero and the corresponding change of energy would be zero. The assertion derived from this perspective dictates that, whenever a field is applied, the spectral emission would split into three lines. Arguably, it has been believed that, there exists a magnetic momentum of _{} emanating from the motion of a given orbital electron. Consequently, Zeeman splitting can be calculated from the Hamiltonian equation below. _{}. The splitting is dependent on the magnetic field orientation to its angular momentum. In the recent studies, it has been found out that, the electrons also possess an associated magnetic momentum_{}, angular momentum J and its orbital momentum. The electron spin has a Zeeman Effect of different magnitude as compared to the orbital angular momentum due to fact that, the spin electron possesses a magnetic momentum that is twice the orbital spin.
_{}
_{} is the Lande gfactor found in the LS coupled atom
The electron energy shift perturbation is exhibited as
_{Having B indicating the orientation of the Z axis}
_{Change in energy would be determined by }
_{}
Where, _{ (2)}
The zeman splitting with the magnetic field in the z direction would be given as
This report considers the Zeeman splitting of mercury atoms of wavelengths 546.1 nm (_{})
Question T1
When the spectral lines are viewed in the plane perpendicular to
, what are the polarisations associated with the
and
transitions?
When , the polarization spectral lines viewed 1/2. It appears to be parallel to the magnetic field. At, the polarization spectral lines is at 3/2 and 2 respectively. They appear to be perpendicular to the magnetic field.
Question T2
Calculate
for the
and
levels and construct an energy level diagram like that given for the 579.1nm line. Note that, since
you must enumerate the possible transitions VERY carefully.
Using equation (1) for the Lande gfactor:
_{J}L^{2s+1} by using the spectroscopic notation _{} , For
_{J}L^{2s+1} using the spectroscopic notation _{} , For
Question T3
Calculate the unsplit and split frequencies for the 546.1nm line assuming a magnetic field of
. What is the resolution in frequency
required to observe all the predicted splittings distinctly?
By using equation (2) and the answers from Q2 above:
_{}_{}
_{}
Substituting the above values into equation (2) gives:
Finally the resolution in frequency,
Question E1
S
how that the phase lag,, between the beams emerging at
& is
.
Question E2
Show that
A =
is called the Airy function.
Question E3
Question E4
Question E5
Experimental Details
Figure 1: Instrumentation of Zeeman Effect Experiment
Apparatus:
Electromagnetic poles Hg lamp
Lens FabryPerot Etalon
Eye piece Polarizer
Vacuum Pump Fluxmeter

Set up the experiment according to Figure 1 without the source.

Turn on the electromagnet.

Calibrate the magnetic field with the fluxmeter at currents 2A, 4A and 6A.

Measure the inner and outer radius of as many rings as possible and thickness and average them with B =0.

Set the polariser parallel to B and then turn up the magnetic field while observing the ring pattern.

Set the polariser perpendicular to B and measure the radii of as many rings as possible for currents of 3A and 6A.

Compare experimental results with theory.
Calibrated magnetic field with fluxmeter at currents of 2, 4, and 6A.Table1:
Current (A) 
Mag Field (T) 
0.149 ± 0.09 

0.302± 0.09 

0.450 ± 0.09 
Figure 1: Calibration of the magnetic field with the standard errors.
Table 2: Measurement of radius and thickness of three rings with B =0.
Inner Rad _{} 
Outer Rad _{} 
Thickness _{} 
Average_{} 

2_{} 

4_{} 

6_{} 

Figure 2: graph of ring radius squared
against ring numberat B=0
Table 3: Ring number and radius at B =0.
_{} 
_{}^{2} 

1.62 x 10^{4} 
2.63 x 10^{8} 

2.19 x 10^{4} 
4.80 x 10^{8} 

2.63 x 10^{4} 
6.95 x 10^{8} 
From the graph above and the following equation
= _{}^{2 }which is constant
. using the following equation
Calculation of Table 4:
_{} 
_{} 
_{}^{2} 

1.62 x 10^{4} 
1.62 x 10^{4} 

2.19 x 10^{4} 
0.57 x 10^{4} 

2.63 x 10^{4} 
0.44 x 10^{4} 
The mean value of
is therefore the finesse of the etalon is using _{}
The minimum resolvable frequency shift for = 546.1nm can be found by this _{} which is
= Hz
Where is d = 6.35 ± 0.15 mm.
R which is the reflectivity of the etalon can be calculated as _{} and gives R=
Table 5: The radii of split rings when the polariser set perpendicular to B for current of 3A.
Current 3.09 ±0.01A, 0.232 T 

Ring Radius _{} 
Average _{} 

2_{} 

114.6667 

154.3333 

4_{} 

190.6667 

216.6667 

6_{} 

Figure 3: graph of ring radius squared
against ring numberat B=0.232 T
Table 6: Ring number and radius at B =0.232 T.
_{} 
_{}^{2} 

1.35 x 10^{4} 
1.82 x 10^{8} 

2.04 x 10^{4} 
4.15 x 10^{8} 

2.52 x 10^{4} 
6.33 x 10^{8} 
The value of Δε can be found by
from the graph above
Δε = 3 · 103 [m].
with the corresponding mean splitting frequency
= Hz
Table 7: The radii of split rings when the polariser set perpendicular to B for current of 6A.
Ring Radii 5.96 ±0.01A, 0.448T 

Ring Radius _{} 
Average _{} 

2_{} 

106.6667 

4_{} 

229.6667 

6_{} 

229.6667 

267.6667 

Figure 4: graph of ring radius squared
against ring numberat B=0.448 T
_{} 
_{}^{2} 

1.43 x 10^{4} 
2.04 x 10^{8} 

2.04 x 10^{4} 
4.16 x 10^{8} 

2.49 x 10^{4} 
6.20 x 10^{8} 
The value of Δ ε = 3 · 103 [m]
with the corresponding mean splitting frequency of 1.5 10^{10} Hz.
Magnetic Field B (T) 
Theoretical Frequency Shift (Hz) 
Experimental Frequency Shift (Hz) 
1.029 x 10^{10} 
0.4 10^{10} 

1.983 x 10^{10} 
1.5 10^{10} 
Analysis and Discussion
The splitting of spectral lines is clearly identified as the Lorentz effect. Whenever the effect of magnetic field is eliminated, a single transition only takes place. From the DP transition. On the other hand the presence of the magnetic field would mean that, the energy associated with the atom would split into 2L +1 components creating a radiating transition occurring between these components. This means_{}. From these observations, there were a total of nine probable transitions. However, only three lines were visible as they possess the same wavelength and energy.
As observed, the inner rings provide a low value in the change of energy as compared to the outer rings. Observing the _{}— lines, it is evident that, the amount of transverse increases proportionally with the increase in the strength of the magnetic field. Constructive interference occurs when _{} . When the lenses with focal length of are brought to focus, bright rings are observed. The square of the radius as indicated in table 3 are linearly related and aid in determining _{}
Consequently, when the pattern of the ring is properly defined, the scale is set to coincide with the fourth ring pattern. There are some systematic errors observed which occur emanate from the uncertainty of 0.2 Amp and a corresponding magnetic field of 0.03kG.
Data obtained from the FabryPerot indicate the transmission coefficient depends on the value of 2. Overtly, the error occurring in the electron spin emanates from the assumption of the electron spin that was not considered during the experiment. This makes the value of S=0. However, in real situations, the spin magnetic strength dipole has to be in the same order as the angular momentum of that magnitude as given by the equation
A factor of about1/2 in splitting difference was obtained which is in tandem with the expected value 2.335_{}
The statistical error in the experiment emanated as a result of fitting with the Lorentzians in relation to the radial correlation rings. However, this error is of insignificant effect as is of very small pixels. Additionally, another error is due to the calculation of the slope. This emanates when the geometric mean is calculated when summing up the _{}.
From the experiment, it is evident that, there was unresolved peaks indicated in the lines of fall occurring between the _{} The effective g factor obtained was 1.745 indicating an anomalous Zeeman Effect. Furthermore, as the magnetic field increases with increase in current, the _{}lines tend to broaden. Contrary, the transition from 7s6s 3s1 to 6p6s 3po are seen to be fully resolved that agrees with the theoretical prediction of _{}
The systematic error was dependant on the experimental analysis as the uncertainty thickness of the FabryPerot was about 7 %. This value explains for the little differences that emerge from the experimental results and the theoretical values. Other errors emerged from the perturbations of the images, persistent noise emanating from the CCD images.
From figure 7, it is evident that, the splitting that is associated with the Zeeman effect that occurs between two energy states was 7.6930.72_{} From the theoretical value, there exists a correlation with the accepted value of 2.345_{} considering the Lande g factor.
Conclusion
There is need for additional measurements to the go and the g1 to improve the performance of the experiment. This would require additional calculations by aid of energy level diagram. A colour filter could be added so that the transition of the Zeeman Effect to another transition is easily observed. The results indicate the conformity with the expected values of the theoretical values.
References
Kenyon, R. (2011). The light fantastic: A modern introduction to classical and quantum Optics. Oxford: Oxford UP.
Zettili, N. (2001) Quantum mechanics: Concepts and applications. Chichester: Wiley.