Zeeman Effect Essay Example

  • Category:
    Physics
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    Assignment
  • Level:
    Masters
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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 (Zeeman Effect) 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 Zeeman Effect 1 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 Zeeman-Lorentz 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 Zeeman Effect 2 emanating from the motion of a given orbital electron. Consequently, Zeeman splitting can be calculated from the Hamiltonian equation below. Zeeman Effect 3. 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 momentumZeeman Effect 4, 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.

Zeeman Effect 5Zeeman Effect 6

Zeeman Effect 7 is the Lande g-factor found in the L-S coupled atom

Zeeman Effect 8

The electron energy shift perturbation is exhibited as

Zeeman Effect 9Zeeman Effect 10Zeeman Effect 11

Having B indicating the orientation of the Z axis

Change in energy would be determined by

Zeeman Effect 12

Zeeman Effect 13Where, (2)

The zeman splitting with the magnetic field in the z direction would be given as

Zeeman Effect 14

This report considers the Zeeman splitting of mercury atoms of wavelengths 546.1 nm (Zeeman Effect 15)

Question T1

When the spectral lines are viewed in the plane perpendicular to
Zeeman Effect 16
, what are the polarisations associated with the
Zeeman Effect 17
and
Zeeman Effect 18
transitions?

WhenZeeman Effect 19 , the polarization spectral lines viewed -1/2. It appears to be parallel to the magnetic field. AtZeeman Effect 20, the polarization spectral lines is at 3/2 and -2 respectively. They appear to be perpendicular to the magnetic field.

Question T2

Calculate
Zeeman Effect 21
for the
Zeeman Effect 22
and
Zeeman Effect 23
levels and construct an energy level diagram like that given for the 579.1nm line. Note that, since
Zeeman Effect 24
you must enumerate the possible transitions VERY carefully.

Using equation (1) for the Lande g-factor:

Zeeman Effect 25

JL2s+1 by using the spectroscopic notation Zeeman Effect 27 , Zeeman Effect 26For

Zeeman Effect 28

JL2s+1 using the spectroscopic notation Zeeman Effect 30 , Zeeman Effect 29For

Zeeman Effect 31

Question T3

Calculate the unsplit and split frequencies for the 546.1nm line assuming a magnetic field of
Zeeman Effect 32
. What is the resolution in frequency
Zeeman Effect 33
required to observe all the predicted splittings distinctly?

Zeeman Effect 34

Zeeman Effect 35

By using equation (2) and the answers from Q2 above:

Zeeman Effect 36

Zeeman Effect 37

Zeeman Effect 38

Zeeman Effect 39

Substituting the above values into equation (2) gives:

Zeeman Effect 40

Finally the resolution in frequency,

Zeeman Effect 41

Question E1

SZeeman Effect 42
how that the phase lag,
Zeeman Effect 43, between the beams emerging at
Zeeman Effect 44 & Zeeman Effect 45
is
Zeeman Effect 46 .

Question E2

Show that

Zeeman Effect 47

A =Zeeman Effect 48

is called the Airy function.

Question E3

Question E4

Question E5

Experimental DetailsZeeman Effect 49
Zeeman Effect 50

Figure 1: Instrumentation of Zeeman Effect Experiment

Apparatus:

Electromagnetic poles Hg lamp

Lens Fabry-Perot Etalon

Eye piece Polarizer

Vacuum Pump Fluxmeter

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

  2. Turn on the electromagnet.

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

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

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

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

  7. 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

Zeeman Effect 51

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 Zeeman Effect 52

Outer Rad Zeeman Effect 53

Thickness Zeeman Effect 54

AverageZeeman Effect 55

2Zeeman Effect 56

4Zeeman Effect 57

6Zeeman Effect 58

Zeeman Effect 59

Figure 2: graph of ring radius squared
Zeeman Effect 60
against ring numberZeeman Effect 61at B=0

Table 3: Ring number and radius at B =0.

Zeeman Effect 62

Zeeman Effect 63Zeeman Effect 64

Zeeman Effect 65Zeeman Effect 662

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 Zeeman Effect 67

Zeeman Effect 68= Zeeman Effect 692 which is constant

.Zeeman Effect 71 using the following equationZeeman Effect 70
Calculation of Table 4:

Zeeman Effect 72

Zeeman Effect 73Zeeman Effect 74

Zeeman Effect 75Zeeman Effect 76

Zeeman Effect 77Zeeman Effect 78-2

Zeeman Effect 79

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 Zeeman Effect 80
is therefore the finesse of the etalon is using Zeeman Effect 81

The minimum resolvable frequency shift for  = 546.1nm can be found by this Zeeman Effect 82 which is
Zeeman Effect 83 = Hz

Where is d = 6.35 ± 0.15 mm.

R which is the reflectivity of the etalon can be calculated as Zeeman Effect 84 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 Zeeman Effect 85

Average Zeeman Effect 86

2Zeeman Effect 87

114.6667

154.3333

4Zeeman Effect 88

190.6667

216.6667

6Zeeman Effect 89

Zeeman Effect 90

Figure 3: graph of ring radius squared
Zeeman Effect 91
against ring numberZeeman Effect 92at B=0.232 T

Table 6: Ring number and radius at B =0.232 T.

Zeeman Effect 93

Zeeman Effect 94Zeeman Effect 95

Zeeman Effect 96Zeeman Effect 972

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 Zeeman Effect 98
from the graph above

Δε = 3 · 10-3 [m].
with the corresponding mean splitting frequency Zeeman Effect 99
= 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 Zeeman Effect 100

Average Zeeman Effect 101

2Zeeman Effect 102

106.6667

4Zeeman Effect 103

229.6667

6Zeeman Effect 104

229.6667

267.6667

Zeeman Effect 105

Figure 4: graph of ring radius squared
Zeeman Effect 106
against ring numberZeeman Effect 107at B=0.448 T

Zeeman Effect 108

Zeeman Effect 109Zeeman Effect 110

Zeeman Effect 111Zeeman Effect 1122

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 · 10-3 [m]
with the corresponding mean splitting frequency of 1.5Zeeman Effect 113 1010 Hz.

Magnetic Field B (T)

Theoretical Frequency Shift (Hz)

Experimental Frequency Shift (Hz)

1.029 x 1010

0.4Zeeman Effect 114 1010

1.983 x 1010

1.5Zeeman Effect 115 1010

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 DZeeman Effect 116P 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 meansZeeman Effect 117. 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 Zeeman Effect 118— 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 Zeeman Effect 119 . 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 Zeeman Effect 120

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 Fabry-Perot indicate the transmission coefficient depends on the value of 2Zeeman Effect 121. 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

Zeeman Effect 122

A factor of about1/2 in splitting difference was obtained which is in tandem with the expected value 2.335Zeeman Effect 123

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 Zeeman Effect 124.

From the experiment, it is evident that, there was unresolved peaks indicated in the lines of fall occurring between the Zeeman Effect 125 The effective g factor obtained was 1.745 indicating an anomalous Zeeman Effect. Furthermore, as the magnetic field increases with increase in current, the Zeeman Effect 126lines 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 Zeeman Effect 127

The systematic error was dependant on the experimental analysis as the uncertainty thickness of the Fabry-Perot 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 7.693Zeeman Effect 1280.72Zeeman Effect 129 From the theoretical value, there exists a correlation with the accepted value of 2.345Zeeman Effect 130 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.