NANYANG TECHNOLOGICAL UNIVERSITY FIRST YEAR COMMON ENGINEERING

FORMAL LAB REPORT

EXPERIMENT S1: EMISSION SPECTROSCOPY AND THE ZEEMAN EFFECT

DATE OF LAB: AUGUST 28, 2002 CLASS: BL15 - LAB1A

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Abstract This experiment is an effort to investigate the emission spectrum of sodium and the “Zeeman effect”. This experiment bases itself on both theories, classical and quantum, and is divided into main sections, emission spectroscopy and the “Zeeman effect”. This experiment uses the diffraction spectrometer to determine the wavelengths of the sodium (Na) spectrum, and compare it with known values. The diffraction spectrometer is calibrated by performing a trial run with the well-known spectrum of helium (He). In addition, the “Zeeman effect” on the splitting of red cadmium spectral lines is observed through the use of the Fabry-Perot interferometer, showing the relationship between magnetic flux densities and spectral line splitting. In all sections, the results of the experiment were found to be positive. The experimental values or observations are close to known values and/or in agreement with theory. The calibration using the helium (He) spectrum proves useful in removing the bias errors, and thus increases the experiment’s accuracy. Through the proper use of calibration and other techniques used in this experiment, determination of the sodium spectrum is accurately found.

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Table of Contents

ABSTRACT .............................................................................................................................. 1

TABLE OF CONTENTS ....................................................................................................... 2

1. INTRODUCTION .............................................................................................................. 3

1.1 OBJECTIVES ...................................................................................................................... 3 1.2 THEORY ............................................................................................................................ 3

1.2A Diffraction and the Emission Spectrum ............................................................................... 3 1.2B Spectroscopy.................................................................................................................... 4 1.2C The Diffraction Grating ................................................................................................... 4 1.2D Theory of Atomic Spectra ................................................................................................. 6 1.2E Theory of the Zeeman Effect .............................................................................................. 8

2. METHOD AND MATERIALS........................................................................................ 10

2.1 APPARATUS ..................................................................................................................... 11 2.2 PROCEDURE.................................................................................................................... 11

Part 1 Calibration Using the He Spectrum ................................................................................ 11 Part 2 Characteristic Wavelengths of the Na Spectrum ................................................................. 12 Part 3 Observing the “normal Zeeman effect”.............................................................................. 12

3. RESULTS ........................................................................................................................... 13

3.1 DATA/OBSERVATIONS ................................................................................................... 13 3.2 CALCULATIONS ............................................................................................................... 15

Part 1 Calibration Using the He Spectrum ................................................................................ 15 Part 2 Characteristic Wavelengths of the Na Spectrum ................................................................. 17

4. DISCUSSION AND CONCLUSION.............................................................................. 19

Part 1 Calibration Using the He Spectrum ................................................................................ 19 Part 2 Characteristic Wavelengths of the Na Spectrum ................................................................. 20 Part 3 Observing the “normal Zeeman effect”.............................................................................. 22

APPENDIX A (STATISTICAL METHODS)..................................................................... 23

APPENDIX B (TABLES) ..................................................................................................... 25

APPENDIX C (THE FABRY-PEROT INTERFEROMETER) ....................................... 26

REFERENCES ...................................................................................................................... 27

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1. Introduction Emission spectroscopy is the study of the composition of light that a gas emits when subjected to an electric discharge. Each element produces a unique light pattern known as the emission spectrum due to their distinct electron orbits. The second part of this experiment will study this emission spectrum using a diffraction grating spectrometer. Since direct observation of light beyond the visible spectra is not possible, the experiment will be limited on determining spectra between 400-700nm (the visible spectrum). But firstly, helium, whose emission spectrum is well-known, will be used to experimentally determine the unique characteristic of the spectroscopy apparatus, or in the technical sense calibrate it. In turn, the calibrated spectroscope is then used to observe sodium’s spectrum. Sodium has an interesting spectrum in the fact that over 95 percent of its luminosity comes from two bright wavelengths of light in its spectrum known as the sodium D-lines. These two lines are moreover, less than 1 nm in wavelength difference, and thus difficult to observe. In the 2nd order spectrum, however, the spectral lines will show more separation, and so through this spectrum will the distinct D-lines be observed. The third part of the experiment is an observation of the “Zeeman effect” on a cadmium (Cd) spectral lamp. This effect, discovered by Nobel Laureate Pieter Zeeman by his researches into the influence of magnetism on radiation phenomena, reveals the mechanism of light radiation and on the nature of matter and electricity. In this experiment, the excited cadmium atoms will be subjected to a varying magnetic field created by an electromagnetic coil. The effect will be qualitatively observed through a Fabry-Perot interferometer.

1.1 Objectives The objectives of the experiment are (1) to use the well-known spectral lines of helium (He) to calibrate the diffraction grating spectrometer and (2) determine the grating constant. (3) Then use the calibrated spectrometer to determine the wavelengths of the spectral lines of sodium (Na), and evaluate it with accepted values. The observation of the sodium spectrum will also include (4) the determination of its fine structure splitting in which the difference between the two D-line wavelengths will be determined. Lastly, (5) the "Zeeman effect" on the cadmium spectrum is to be observed through the Fabry-Perot interferometer, and thus a relationship between the increasing magnetic flux density and splitting will be formed.

1.2 Theory This section will provide more insight into the theory behind the experiment. The theoretical explanation will elaborate on the nature of light, the process of diffraction, and the “Zeeman effect”. In addition the necessary derivation of formulae will be presented and discussed with visual aids.

1.2A Diffraction and the Emission Spectrum

Light is ordinary, yet interesting and complicated. It cannot be classified as either only a particle or a wave. When considering visible light or generally all electromagnetic disturbances, one should consider it as both a particle and wave. Like a coin that cannot be regarded as either heads or tails, but instead a two-sided entity, so must light be considered as a duality between a particle and wave. Some experiments reveal light’s wave-like properties, while others better show its particle nature.

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The experiment performed in this particular lab draws on both aspects of light. When light travels through relatively narrow slits, small apertures, around obstacles or sharp edges, it diverges. This wave-like property of light is known as diffraction. Like any mechanical wave, electromagnetic wave sources can diffract and interfere with each other constructively and deconstructively with light of different wavelengths diffracting at different angles. By using this property, we can find the composition of light from a source by diffracting it into its divided spectrum. This spectrum of light is known as an element’s emission spectrum, or the spectra of light that it emits when its atoms are excited. The simplest line spectrum is that for the atomic hydrogen since its atomic structure is most simple (one proton and one electron). Other atoms exhibit completely different line spectra. Because no two elements have the same line spectrum, this phenomenon represents a practical and sensitive technique for identifying the elements present in unknown samples. This process is often used by scientists to determine the elemental make-up of luminous objects, like distant starts and galaxies.

1.2B Spectroscopy

All objects emit radiation characterized by a continuous distribution of wavelengths. In sharp contrast to this continuous distribution spectrum is the discrete and unique line spectrum emitted by a low-pressure gas subject an electric discharge (Electric discharge occurs when the gas is subjected to a potential difference that created an electric field larger than the dielectric strength of the gas therefore ionizing it). The visible radiation is actually, similar to that of white sunlight, a composite of various wavelengths of light, although not as rich and continuous as that of sunlight. When light from a gas discharge is examined with a spectroscope, it will be found to consist of a few bright lines of color on a generally dark background. Observation and analysis of this emitted light is known as emission spectroscopy, the process used in this experiment.

1.2C The Diffraction Grating

There are several methods that can be used to separate a light source into its component wavelengths and produce the desired characteristic spectrum. In this experiment, a transmission diffraction grating is used to produce spectra from gas discharge tubes of helium and sodium. The grating uses the property of light: radiation of different wavelengths diffract at different magnitudes (the shorter the wavelength the smaller the angle of diffraction). After light from the source is focuses through the collimator tube, it passes through a diffraction grating (refer to diagram 1.2a). The diffraction grating is a useful device for analyzing light sources. It usually consists of a large number (over 500 lines/mm) of equally spaced parallel slits, and is transparent in appearance. The spaces between the lines are transparent to light and hence act as separate slits or sources. The distance between two consecutive lines is called the grating constant d, and is usually manufactured in the range of 1000 to 2000 nm.

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DIAGRAM 1.2A (Diffraction and Interference)

When light of wavelength λ propagates through a diffraction grating of spacing d, it is diffracted. Due to light’s wave-like properties, the slits of the grating will create an interference pattern in the resulting projection of spectra. There will be areas of maximum constructive interference and dark areas of maximum deconstructive interference. This pattern continues indefinitely, but each successive image becomes fainter (Diagram 1.2A): the direct image with no interference is known as the principle image, the first array of spectrum to either the left or right of this principle image is called the 1st order spectrum, the next being the 2nd order spectrum, etc. Inspection of the line spectra will be where the intensity is at a maximum and thus easiest to observe. Such maxima intensities are produced if the angle of diffraction θn satisfies the following condition:

nλ = d sinθn (Equation 1.2) where n is an integer (0,1,2,3…) called the order number and corresponds to the order of the spectrum being observed (refer to diagram 1.2a for a visual explanation). This relationship can thus determine the wavelength of the various lines assuming that d and the angle θn are known. A schematic drawing of the spectroscope used to measure angles in a diffraction pattern is shown in Diagram 1.2b. Light that exits the collimator tube and strikes the diffraction grating undergoes diffraction. The component wavelengths of the spectrum leave at angles that satisfy Equation 1.2. The telescope tube is used to view the image of the slit. The wavelength can be determined by precisely measuring the angles of the spectral images.

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DIAGRAM 1.2b (Spectroscope)

1.2D Theory of Atomic Spectra

In a different approach to understanding this phenomenon, we will turn to a quantum view point. Niels Bohr’s research into the structure of the atom during the early 20th century revealed theories that this experiment is rooted, and therefore important to understand. His model contained classical as well as other revolutionary postulates (now known as quantum physics), but the general idea as it applies to basic atoms is as follows:

(i) The electron moves in circular orbits around the proton nucleus under the influence of the Coulomb force of attraction.

(ii) Only certain electron orbits prevent the electron from leaking energy in the form of radiation, and so are stable. The size of the allowed orbits are determined by conditions imposed on its orbital angular momentum and are fixed distances from the nucleus.

(iii) Radiation is emitted by the atom when the electron ‘jumps’ from a more energetic orbit to a lower-energy orbit. (refer to Diagram 1.2C)

DIAGRAM 1.2C (Bohr’s Orbital Shells)

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In addition, it is also known from the combined works of Einstein and Planck, that light comes in discrete packets known as photons whose energies are proportional to their frequency f in the expression:

E1 – E0 = hf where h is Planck’s constant (h = 6.626 x 10-34 Js). E1 and E0 are the excited and original state of the electron respectively. Additional mathematical manipulations between the equation for the electrons angular momentum and the equation for the electron’s kinetic energy, yields the equations for allowed orbital radii. A quantized radius in turn leads to the conclusion of quantized energy levels as well. The specific equations for each element differ according to different levels of angular momentum degeneracy, quantum defect, and effects of other quantum numbers. The complex nature of how electrons encircle a nucleus in three dimensions makes formulating these equations relatively difficult. The details of these relationships are beyond the scope of this brief theory overview and whose details are not as important. Since this experiment examines the sodium spectrum, it is most sensible that we focus on the formula for the energy states of sodium. The formula is as follows: Enl = mee4 (Znl)2 1 8ħ2 n2 where Enl is the energy of any particular orbital (dependent of n and l which are the principle and orbital quantum numbers respectively); Znl is the effective atomic number, me is the electron mass, e is the electron charge, and ħ is h/2π. An approximation of this formula yields: Enl = mee4 1 _ 8ħ2 (n-µnl)2 where µnl is the quantum defect that depends on some slight extent of n and decreases as l increases. The interaction of the spin angular momentum S (a result of the spin magnetic quantum number s) of the electron with its orbital momentum L gives rise to a reduction in the degeneracy of the total angular momentum. With this proper understanding of energy levels it is possible, through perturbation theory, to surface a quantum prediction of the possible spectrum of sodium.

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DIAGRAM 1.2D (Sodium Energy Levels)

Scientists studying the behavior of atoms had earlier discovered the so-called ``fine-structure'' of spectra, where one spectral line under higher resolution splits into two or more spectral lines; and so on, and so on, for thousands of journal pages. The same applies sodium, in which the fine-splitting can be observed under its 2nd order spectrum. This doublet is known as the Sodium D-lines and has an experimentally accepted value of 588.9950 and 589.5924 nanometers. Take note from analyzing Diagram 1.2D that the sodium spectrum is dominated by the bright doublet. It is evident that these lines are created from electron transitions from the 3p to the 3s levels. The line at 589.0 has twice the intensity of the line at 589.6 nm. Taking the range from 400-700nm as the nominal visible range, the strongest visible line second to the D-lines is the line at 568.8205 which has an intensity of only about 0.7% of that of the strongest line. All other lines are a factor of two or fainter than that one, so for most practical purposes, all the light from luminous sodium comes from the D-lines. Because the two sodium D-lines have such similar values for their wavelength, it is difficult to distinguish them while observing through the spectroscope. However, the higher the spectrum image order the more the separation between the lines will increase. Thus it is vital to examine the second order spectrum for sodium if the individual D-lines are to be observed at all.

1.2E Theory of the Zeeman Effect

By the late years of the 19th century, scientists were investigating into the nature of light, continuing the earlier works of Faraday and Maxwell. Researches were pointing towards a new theory of light. In 1896, through the joint efforts of Lorentz and Zeeman, it was found, where Faraday failed, that

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magnetic fields do in fact affect the spectral lines of cadmium and generally any spectra of light. The spectral lines apparently split into three distinct lines when the magnetic field was sufficiently intense. The splitting up of these spectral lines of atoms within the magnetic field was thereafter known as the Zeeman Effect. This simple splitting up of one spectral line into three components is called the "Normal Zeeman Effect". This phenomenon cannot be explained in classical terms using waves, but instead through quantum theory. Further research into this phenomenon led to the quantum discovery of the existence of magnetic quantum numbers. An orbiting electron can be considered an effective current loop with a corresponding magnetic moment. Magnetic interaction occurs when this moment created by the electron loop is placed in an external magnetic field. The fields interact and affect the electron’s momentum, but the changes in the electron’s orbital angular momentum, like all other quantum effects, are quantized. The quantization of the direction of L with respect to an external magnetic field is referred to as space quantization. The additional energy levels provided by the orbital magnetic quantum numbers give rise to the observed "Zeeman effect". The splitting of the cadmium (Cd) spectral line λ = 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 transition from orbital D P producing a wavelength of 643.8 nm, as indicated by Figure 1.2E. DIAGRAM 1.2E (Orbital Magnetic Numbers on Energy Levels)

In the presence of a magnetic field the associated energy levels split into 2L + 1 components. The possible radiating transitions between these components are ∆ML = +1, 0, -1, where ∆ML is the difference between the magnetic quantum numbers. Take for example in Diagram 1.2E, the drop from +2 to +1. The ∆ML is identical to a drop from +1 to 0, which is also 0. Although there are a total of nine permitted transitions, there are only three possible combinations of different energies. The other combinations yield drops that have the same energy and hence the same wavelength. Therefore, only three distinguishable lines will be visible.

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The effects of the magnetic field makes some states move up in energy, and some move down. Reduced symmetry has lead to reduced degeneracy. Transitions that formerly had the same change in energy now have slightly different. For the radiating electrons, a change in energy in the presence of a magnetic field can be related to the difference in wave numbers of one of the labeled σ-lines to the central line: ∆E = hc(∆v/2) In an analysis of the spin magnetic quantum number, we formulate another formula to express the change in energy ∆E, as it is proportional to the magnetic flux density B: ∆E = µbB where µb is known as Bohr’s magneton (µb = eħ/2me). Hence by combining the two equations, the following relationship between the difference of wave numbers and the magnetic flux density B can be obtained: (∆v/2) = µbB/(hc) The equation shows the proportionality of (∆v/2) to B and by graphing this relationship, the value of Bohr’s magneton µb can be experimentally determined. FIGURE 1.2F (Zeeman splitting as function of magnetic flux density)

Thus from quantum analysis, the Zeeman splitting of spectral lines is predicted and quantitatively related through Figure 1.2F.

2. Method and Materials The procedures and apparatus for the two parts of the experiment have been separated the three parts and listed accordingly.

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2.1 Apparatus Part 1 & 2

- Spectrometer - Diffraction grating, 590 lines/mm (d = 1684 nm) - Helium spectral lamp - Sodium discharge tube - Power supply for spectral lamps - Lamp holder for spectral lamps - Tripod base

Part 3 (See Appendix C for additional details)

- Cadmium lamp and power supply - Electromagnet with pole shoes and rotating table - Variable transformer, 25VAC/20VDC, 12A - Electrolytic capacitor, 22000µF - Optical profile bench with adjustable base, lenses and lens holder - Iris Diaphragm, polarizing filter and quarter-wave plate (mica): A circular device with a

variable diameter, used to regulate the amount of light admitted to a lens. - Spirit level - Digital Teslameter: Measures magnetic flux density - Hall probes - Fabry-Perot interferometer

2.2 Procedure This section will provide an elaborate step-by-step account of the procedures carried out during the course of the experiment.

Part 1 Calibration Using the He Spectrum

(1) The equipment for Part 1 was set up accordingly. The vernier and grating were adjusted according to their operating instructions. The diffraction grating was placed in its holder on the spectrometer as shown in Diagram 1.2b. Necessary precautions were taken to ensure that the grating did not move through the course of the experiment.

(2) The helium discharge tube was placed in the lamp holder. For safety reasons, additional caution was taken to switch off the power supply when connecting the discharge tube. It was also advisable to NOT contact with the high voltage electrodes of the lamp.

(3) Once the helium lamp was turned on, the spectrum tube was moved as close as possible to the slit in the collimator tube. The lamp was left on for an additional 5 minutes to allow it to warm up to its full illuminating power. The lamp housing was adjusted so that air could circulate freely through the ventilation slits and thus not cause overheating.

(4) The telescope was aligned with the collimator tube of the spectrometer. The slit in the collimator and the eyepiece of the telescope was adjusted until a sharp triangular image of the slit is obtained. The vertical cross hair of the telescope must also be in focus and in the field of view.

(5) The telescope could be locked through the means of the knurled head screw below the spectrometer table. The goniometer was finely turned until the vertical crosshair of the telescope was at the left tip of the image. Using the mounted magnifying lens and vernier, the

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initial angle was recorded from the graduated circular scale of the spectrometer with an accuracy of 1 arcminute.

(6) The telescope was unlocked and moved to the left of the spectrometer table until images of all the spectral lines for helium as indicted in Table B of Appendix B were located.

(7) Using the vernier scale and the goniometer at each locked position of the telescope, careful measurements to the nearest arcminute of the 1st order angles were taken for each wavelength of helium. Each measurement was made with the crosshair positioned at the top left tip of the image. These measurements were repeated for the right side of the spectrometer as well.

(8) The angles θL and θR for each of the wavelengths were recorded in Data Table 1.

Part 2 Characteristic Wavelengths of the Na Spectrum

(1) Without displacing the diffraction grating, the spectrum tube power supply was switched off, and the helium discharge tube was allowed to cool down to room temperature. Carefully, it was replaced with the sodium discharge tube.

(2) The power supply was again switched on and the discharge tube was placed as close as possible to the slit. The same precautions were taken with the sodium tube as was with the helium. The lamp was allowed to warm up for 5 minutes to attain its full illuminating power.

(3) The telescope tube was rotated back to the initial angle and carefully adjustments to the position of the Na discharge tube were made until a sharp image of the slit was seen directly through the grating.

(4) Careful measurements to the accuracy of 1 arcminute were made for the first five wavelengths of the visible 1st order sodium spectrum on both the sides of the initial angle.

(5) The two angles θL and θR for each wavelength was recorded in Data Table 2. (6) To observe the sodium D-line split, the spectrometer was set for the red line in the 2nd order

spectrum with the telescope farther left or right of the spectrometer table. (7) With the telescope locked, the goniometer was used to align the vertical cross hair in the

telescope with the red line in the 2nd order spectrum. The spectrometer table was then unlocked by turning the knurled head screw under the grating table and the spectrometer scale 0 was realigned with the vernier scale 0. The alignment was checked with the magnifying lens and finally the spectrometer table was locked. These processes were carefully done not to displace the diffraction grating.

(8) Using the goniometer, the vertical cross hair was aligned with the longer wavelength sodium D-line and then with the shorter-wavelength sodium D-line. At these two positions, recordings were made of the readings for θL1 and θL2 with the accuracy of 1 arcminute. Measurements were repeated for θR1 and θR2 with the telescope on the other side of the spectrometer table.

Part 3 Observing the “normal Zeeman effect”

(1) The Fabry-Perot interferometer was set up as shown in Figure 2 for the observation of the σ-lines of the transverse Zeeman effect.

(2) Without any application of a magnetic field, the ring pattern of the cadmium (Cd) spectral lines was observed through the lens.

(3) The coil current of the electromagnet were increased in intervals of 3 Amps: 3, 6, and 9 A. The corresponding magnetic flux densities at these currents were recorded using the digital teslameter.

(4) For each interval of increased magnetic flux density due to the electromagnet, the change in the ring pattern was observed.

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3. Results This section comprises of tables of measurements and calculations obtained from the experiment. In addition, the mathematical framework used for the calculations in this experiment is presented at this point.

3.1 Data/Observations The following tables present the angle measurements for the various spectral lines recorded by using the graduated circular scale of the spectrometer. The values were measured by positioning the crosshair on the top left corner of the spectral lines, but what appeared through the spectroscope more like distorted rectangles. These angles are relative to the initial angle θ0 set by positioning the crosshair on the top left corner of the principle image. DATA TABLE 1: Spectrometer angle readings from the He spectrum

He Spectrum ӨL (degrees/minutes)

ӨR (degrees/minutes)

Ө = ½(ӨR-ӨL) (degrees)

Red 24/06 335/51 155.90º

Yellow 21/03 338/57 158.95º

Green 17/54 342/12 162.10º

Greenish blue 17/45 342/03 162.28º

Bluish green 17/10 343/00 163.00º

Blue 16/16 343/54 164.95º

DATA TABLE 2: First order spectrometer readings from the Na spectrum

Na Spectrum ӨL (degrees/minutes)

ӨR (degrees/minutes)

Red 22/18 337/12

Yellow 21/12 338/15

Yellowish green 20/30 339/0

Green 1 18/24 341/6

Green 2 17/45 341/48

This data table was recorded during the second part of experiment when the D-lines were observed from the 2nd order spectrum. The two lines were so near that in fact even through observing them from the 2nd order spectrum, they overlapped each other. The recordings were approximations of the difference in the position of their top left corners. DATA TABLE 3: Spectrometer readings from observing the sodium D-line split

Measurements with respect to the red line in the 2nd order spectrum and spectrometer scale 0 initially in line with vernier scale 0

Left side of spectrometer

(degrees/minutes)

Right side of spectrometer ӨR

(degrees/minutes)

ӨL1 ӨR1 Position of telescope with cross hair in line with the longer wavelength Na D-Line 45/12 312/6

ӨL2 ӨR2 Position of telescope with cross hair in line with the shorter wavelength Na D-Line 45/6 312/12

Observations of the Zeeman Effect

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In Part 3 of the experiment, the “Zeeman effect” was observed. Through the lens of the Fabry-Perot interferometer, the normal red spectrum of cadmium could be seen. They formed numerous red circles on a black background with the larger surrounding the smaller in an endless pattern. Although there was no reproduction of the images seen through the interferometer, a diagram that approximates the images can be referred to below (Figure 3.1A). As the current through the electromagnet and consequently the magnetic flux density increases, the spectrum gradually broke apart into three distinct lines. At 3 A, the spectrum was seen to thicken, but no distinct lines were prominent. When the current was increased to 6 A, there revealed two visible lines. When the current was set at the maximum at 9 A, the Lorentz-triplets were clearly discrete. FIGURE 3.1A (Splitting of Cadmium Spectral Lines)

Figure 3.1B: The relationship between the increasing coil current and the magnetic flux density as sampled using the digital teslameter.

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3.2 Calculations Due to the numerous steps involved, the calculations have been broken down into two sections Part 1 and Part 2 for clarity.

Part 1 Calibration Using the He Spectrum

The angles in degrees and minutes must first be converted into degrees for further calculations. Then to obtain θ, the average of the angles on the left θL and right θR must be taken. A sample calculation for red: Example Calculation 3.28 θ L = 24º 06’ = (24 + 6/60) degrees = 24.1º θ R = 335º 55’ = (335 + 51/60) degrees = 335.85º Example Calculation 3.27 θ = ½ | θ R - θ L| θ = ½ | 335.85 – 24.1| θ = 155.875 ≈ 155.9º Then using the relationship of Eq.1.2, the grating constant each particular angle yields can be found. Sample for red: Example Calculation 3.26 nλ = d sinθn (since the 1st order spectrum is being observed, n = 1) λ = d sinθn

d = λ/sinθn (the value for the corresponding wavelength λ can be found in the table of accepted wavelengths for helium in Appendix B)

d = 667.8/(sin 155.9) d = 1635.4401 ≈ 1635.4 nm The mean grating constant can be simply calculated by taking the average of all the grating constants (i.e. (1635.4 + 1635.9 + 1632.0 ….. )/6 or see Appendix A). CALCULATIONS TABLE 1: Calculating the grating constant d and its standard error from the He spectrum

He Spectrum Ө = ½ |ӨR - ӨL| (degrees)

Grating Constant d (nm)

Mean Grating Constant đ (nm)

Standard Error* of đ

α (nm) Red 155.90 1635.4

Yellow 158.95 1635.9

Green 162.10 1632.0 1624.9 4.3579

Greenish blue 162.28 1617.1

Bluish green 163.00 1612.0

Blue 163.95 1617.1

*See Appendix A for statistical methods in calculating the standard deviation and standard error. In this distribution the standard deviation was 10.6747.

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The next step is to calculate sin θ for each angle. This calculation is pretty much straight-forward and the results are recorded in Calculations Table 2. The two other columns, wavelengths and the angle θ are identical and taken from those from Calculations Table 1. This data will be used for the plotting of the calibration curve. CALCULATIONS TABLE 2: Data points from the Helium spectrum for the calibration curve

He Spectrum Wavelength* (nm)

Ө = ½ |ӨR - ӨL| (degrees)

Sin Ө

Red 667.8 155.90 0.40833

Yellow 587.6 158.95 0.35918

Green 501.6 162.10 0.30736

Greenish blue 492.2 162.28 0.30437

Bluish green 471.3 163.00 0.29237

Blue 447.1 163.95 0.27648

* Values for the helium spectral wavelengths are taken from the accepted-value figures in Appendix B. Using the calculations from Calculations Table 2, a calibration curve could be drawn as shown in Figure 3.2A. The graph plots sin θ along the x-axis and the corresponding helium wavelengths λ along the y-axis. A linear least-squares fit regression has been fitted to the graph.

y = mx + b y = 1684.7x – 19.05 The mathematics to find the values of m and b for the linear equation are relatively tedious and lengthy, thus are not included in this section, although it may be referred to in Appendix A. The correlation coefficient (r=1.000) indicates a strong correlation between these two variables and denotes a strong integrity of the regression. In observation of the procedure and apparatus, it is noted that the graduated circular scale was in spaced every 0.1 degrees (6 arcminutes), and thus it is highly possible that an estimation error could occurred within this interval. Accordingly, error bars on the y-axis (sinθ) have been included to compensate for this imprecision (sin 0.1 ≈ ± 0.001745). Within this margin of error, the regression accounts for all the data points. Because of the relationship đ sinθ = λ, the slope of the curve, which is λ/sinθ, is equal to d. Hence the grating constant d can be calculated from this least squares fit. d = m dc = 1684.7 nm When we compare the grating constant dc to the mean grating constant đ, we obtain their relative discrepancies:

Percent error = |E – K| x 100 (EK)aver = |1623.7 – 1684.7| x 100 (1684.7+1623.7)/2 Percent error = 3.69 % The grating constant dc from the calibration curve is notably larger than the mean grating constant đ. In comparison with the manufacturer’s value of d=1684nm, dc appears to be a more accurate result.

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The same calculations for percentage errors can be made with respect to the manufacturer’s value for the grating constant dm (see appendix A).

FIGURE 3.2A (Helium Calibration Curve)

Calculations Table A: Relative percent deviations between the grating constants obtained through

mean value, calibration curve, and manufacturer’s specification. Calibrated grating

constant (dc=1684.7) %

Mean grating constant (đ=1623.7)

% Grating constant dc - 3.688

Grating constant đ 3.688 -

Manufactured grating constant (dm=1684nm)

0.04157 3.581

Part 2 Characteristic Wavelengths of the Na Spectrum

Calculations Table 3 ultimately calculates the wavelength of the Na spectrum. The average is taken in the same manner as was for the helium spectrum (see examples 3.27 and 3.28). The wavelength again can be determined by using the relation of Eq.1.2 (see example 3.26). The value the grating constant d

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used in this calculation was the mean grating constant đ obtained earlier from the helium calibration trials. CALCULATIONS TABLE 3: Calculations of the wavelengths in the Na spectrum and their percentage errors using the 1st order diffraction grating equation

Na Spectrum Ө = ½ |ӨR - ӨL| (degrees)

Sin Ө Wavelength ( λ=đ sinθ)

(nm)

% error in λ*

Red 157.45 0.38349 622.7 0.8911

Yellow 158.53 0.36601 594.4 1.020

Yellowish Green 159.25 0.35429 575.3 1.339

Green 161.35 0.31979 519.2 0.9920

Green 162.03 0.30860 501.1 0.4813

*Please see Appendix A for calculations of percentage error. The calculations done for Calculations Table 4 are similar to those done for Calculations Table 3 with the exception of the value used for the grating constant d. Instead of using the mean grating constant đ, however, the calibration curve was used. As the graph shows, the function y = 1684.7x -19.05 (Equation 3.2) resembles Equation 1.2. λ = d sinθ + b where y = λ, d = m, sinθ = x. However, a new factor b has been introduced. This can be treated as the error factor. It causes the graph’s vertical translation. The bias errors that were present in the experiment are accounted in this constant b. This calibration curve can now be used to calculate the wavelength of sodium. Equation 3.2 is kept in its form. Since the objective is to calculate λ, the x in the equation will vary. Here is a sample calculation for the red wavelength of the sodium spectrum: y = 1684.7x -19.05 (since sinθ = x and y = λ) λ = 1684.7(sinθ) – 19.05 λ = 1684.7(0.38349) – 19.05 λ = 627.0 nm The following table shows the new values of y that correspond to the different values of x along with the percent error relative to accepted values of sodium in Appendix B. CALCULATIONS TABLE 4: Determining the wavelengths in the Na spectrum and their percentage errors using the calibration curve

Na Spectrum Ө = ½ |ӨR - ӨL| (degrees)

Sin Ө Wavelength λ from calibration curve

(λ=đ sinθ + b) (nm)

% error in λ*

Red 157.45 0.38349 627.0 1.588

Yellow 158.53 0.36601 597.6 1.564

Yellowish Green 159.25 0.35429 577.8 1.779

Green 161.35 0.31979 519.7 1.089

Green 162.03 0.30860 500.7 0.4010

*Please see Appendix A for calculations of percentage error.

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The wavelengths of both the shorter and longer D-lines were calculated using Eq.1.2 and the mean grating constant đ was used. The differences between the shorter and longer wavelength were then calculated and compared. The average of the shorter wavelength can be calculated as follows: (575.5 + 601.9)/2 = 588.7 nm The average of the longer wavelength can be calculated similarly: (576.5 + 602.9)/2 = 589.7 nm This method is applied to the right side as well, and produces the results in Calculations Table 5. CALCULATIONS TABLE 5: Calculating the sodium D-line split

Left side of spectrometer

(nm)

Right side of spectrometer

(nm) Shorter wavelength λ1 of sodium D-line 575.5 601.9

Longer wavelength λ2 of sodium D-line 576.5 602.9

Difference (λ2 - λ1) between the longer and the shorter wavelength of the sodium D-line

1.0 1.0

Average of wavelength difference (λ2 - λ1)

1.0

4. Discussion and Conclusion In this section, the results will be discussed and analyzed. Due to the large extent of material covered, this section has also been subdivided into the three experimental parts for organization.

Part 1 Calibration Using the He Spectrum

The calibration using the helium (He) spectrum made the experiment much more accurate and precise. It is apparent that the value of the mean grating constant d deviates somewhat from the manufacture’s specifications, but this is due to many possible biases in the experiment. The calibration curve effectively accounts for this and amazingly produces a constant that is only 0.04% (Calculations Table A) from the manufacturer’s value. Calculations Table 1 reveals a pattern; as the angle of the spectral lines increases, the deviation of the calculated d from the mean value increases. As a personal observation, it was noticed that the spectral lines (or rather distorted squares as viewed through the spectroscope) became increasingly distorted and vertically translated as the telescope tube was moved from the initial angle. This can explain the reason for the increase in deviation as the measured angle increases for it becomes increasingly difficult to judge a “top left corner”. To compensate for the up to 3% error (Calculations Table A), there must also be additional sources of bias elsewhere in the experimental setup, which could include biased inaccuracies of the apparatus used, or a faulty assumption in the experiment. Nonetheless these errors are consistent (Calculations Table A), and so the calculated values of d are relatively consistent with each other with a standard deviation of only 10.67 nm and standard error of 4.3579.

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Shifting our focus to the calibration curve (Figure 3.2A), we find that again the errors are biased, and a least squares regression adequately fit the plotted data (λ vs. sin θ) within the range of the estimated error bars. This indicates that the procedure used was fairly consistent and precise, and the error bias is from some other non-procedural source. The value for the grating constant dc obtained from the calibration curve is remarkably similar to the real value denoted by the manufacturer dm (Calculations Table A). Note that in the calculations leading to dc, the helium (He) wavelength λ was introduced. The grating constant in the calibration curve is calculated directly from the relationship between the wavelength to the sin of the angle θ (d = m = y/x = λ/sinθ). Any possible bias errors would have to generally affect all measured values of θ, which would result in a vertical translation of the graph (Figure 3.2A). In essence such biases affect only the y-intercept of the graph or constant b, but not the slope m, which is the relative relationship between the sin of the angles and the wavelengths. So it comes to a conclusion that the calibration curve yields a more accurate grating constant.

Part 2 Characteristic Wavelengths of the Na Spectrum

Calculation Tables 3 and 4 calculate the different wavelengths of the sodium (Na) spectrum. Both results are within remarkably small margin of error, although it is interesting to note that Table 4, which used the calibration curve, yield slightly higher error percentages. Table 3 produces wavelengths values slightly greater than the accepted values, but Table 4 produces even more strayed results. This is opposite to expectations, since the value of the grating constant dc from the calibration curve is shown to be a more accurate estimation in Part 1. The fact that it yields less accurate results with the sodium (Na) wavelengths is peculiar. This apparent contradiction can possibly be explained by comparing the nature of the two values of d. The fact that the experimental procedure is reasonably consistent and causes minimal error must be emphasized first. The experiment is pretty much static, except for the rotating telescope tube. The only significant procedural errors possible are the human judgment used when positioning the crosshair to the “top left tip” of the spectral line, and the reading of the graduated scale which were marked in intervals of 0.1 degrees. In the case of this particular experiment in which the same person performs all the measurements it can be expected that his or her standards for measurement, if he or she is careful, would more or less be consistent. And although there could be misjudgments, they would account for very little and on average they would cancel each other. On the other hand, if he or she is consistent with his or her bias in judgments, it will too contribute to the bias errors. As a result, virtually all the errors that prevent this experiment from producing realistically perfect results are from the ‘bias errors’. These errors, nevertheless, can easily be removed through an appropriate calibration process. The calibration process has to be performed with known values, so that you can obtain some coefficient k that would account for the bias errors. Since the calibration curve has already been explained, this part will focus only on explaining the nature of the mean grating constant d. In the derivation of đ, there is evidence that whatever bias occurred in sampling the angles would be directly incorporated in the resulting value of đ as đ was calculated by λ/sinθ (λ of which is a correct value). It can be conjectured that the mean grating constant đ is actually the product of the true value dm by some bias error constant k. λ/sinθ = k· dm

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Since, λ is correct, there is no question about the value of the wavelength. However, there is bias error in the value of the angles measured as we have discussed. If the two values were “perfect”, then it would be expected that this result would yield the true value of dm, assuming there was no major manufacturing defect. But because this is not the case, the constant k can be supposed to account for the deviation. And so by using this value of d in further experiments one would also take into account the bias present in the general set up and/or procedure of the experiment. It is also possible that any other errors present (present in the 2nd part but not in the 1st) could through good chance be biased towards yielding the more accurate answer. Nonetheless it must be concluded from the results, but not without doubt, that the wavelength values in Calculations Table 3 are more accurate than those using the calibration curve, and yield excellent estimations with the majority of the errors below 1 percent. Explanation of the Fine Splitting of Sodium D-Lines Referring back to Section 1.2D and the sodium D-Lines, we can see the quantum background as to why these two spectral lines are similar in wavelength. And because of this similarity, it is very difficult to distinguish between the two lines when viewed through the spectroscope. Applications of some mathematical relationships and simple calculations using the known values of the D-Lines will prove this point: nλ = d sinθ (from Eq. 1.2) θ = sin-1(λ/d) (in the case of a 1st order spectrum with n=1) (Equation 4.2) The difference between the two wavelengths of the D-Lines is (see Appendix B): (λL – λS) = (589.592 – 588.995) = 0.597 nm Take a value of d (for the sake of this discussion, any arbitrary value of d between 1000-2000nm can be applied as a demonstration), for example dm (=1684nm), and substituting this back into Equation 4.2 yields: θ = sin-1(0.597/1684) θ = 0.0203 degrees or 1.22 arcminutes. Realizing that the graduated circular angle scale on the spectrometer is in intervals of only 0.1 degrees, and that 1.22 arcminutes is a very small angle to observe, it can be understood why the two D-Lines cannot be distinguished apart and thus observed in the 1st order spectrum. However, if one were to recalculate this theoretical estimation but use n=2, the 2nd order spectrum instead, it would yield: θ = sin-1(2λ/d) θ = sin-1(1.194/1684) θ = 0.0406 degrees or 2.44 arcminutes. Apparently, even this angle is also very small and thus still difficult to observe. Nonetheless, this result is more realistically possible since it is about half of the 0.1 degree scale-interval and so can be potentially measured. With the aid of the fine tuning mechanisms, this difference was large enough for measurements to be made, and calculated as in Calculations Table 5. The calculations show values that are remarkably close to the accepted values.

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Part 3 Observing the “normal Zeeman effect”

There is not much to say about the “Zeeman effect”, as the nature of this effect has been adequately discussed in Section 1.2E. The observations and results confirm the theory’s prediction: an external magnetic field will affect the spectral lines of cadmium. Under no external magnetic influence, the interference pattern consists of single red lines (Figure 3.1A), but as the magnetic flux intensity, which is caused by the current in the electromagnet, increases, the splitting of the interference pattern becomes more clear. Conclusion In an overview, all the experimental objectives were satisfactorily accomplished. The preferred calibration process of the spectroscope by using the known helium (He) spectral lines led to some impressive results in determining the sodium wavelengths. The experimentally-determined spectrum of sodium using the calibrated values of both đ and dc was found to be very accurate and deviate from the accepted value on average by less than 1 percent. The sodium D-Lines were successfully observed, measured, and compared to accepted values in the second order spectrum. Their values and differences are similar to the accepted values as well. As for the “Zeeman effect”, it was also successfully observed and confirmed with theory. This experiment is considered a good example of a procedural setup that can be used to measure and determine the spectrum of simple elements; simple by meaning it has a simple electron structure and thus an orderly spectrum. Slight modifications are recommended, however. The entire procedure should be done by one person, so that his or her personal judgment may consistently influence the results, and thus the calibration will be more effective. It also important to make sure the room is as dark as possible so that the spectral lines are as bright and sharp as possible; this makes identifying the top left corner easier. Lastly, a more appropriate amount of time than the 2 hours taken in this experiment should be allocated to performing the experiment. Since the fine tuning and crosshair positioning is a delicate procedure, rushing will only weaken the precision.

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Appendix A (Statistical Methods) 1. Standard Error α The standard error α is defined by the equation: Equation (1) where σ is the standard deviation from the mean xmean, and N is the number of repeated measurements of some quantity x whose ith value is xi and where i sequences from 1 to N. The standard deviation σ is the average distance from any sample x in the distribution to the mean xmean of the sample, and is defined by the equation: Equation (2) Physically, the standard deviation σ is a measure of the precision of the measurement in the following statistical sense. It gives the probability that the measurements fall within a certain range of the measured mean xmean. The common range to be quoted is the range of one standard deviation as calculated by Equation (2). In a normal distribution, well over half of the samples in that distribution should fall within one standard deviation of the mean, and over 99% by three standard deviations. The mean xmean is given by the equation: Equation (3) Thus, in order to calculate the standard error, first calculate the mean xmean using Equation (3), then the standard deviation σ using Equation (2) and lastly the standard error α using Equation (1). 2. Linear Least Squares Fit In many cases of interest, a linear relationship between two variables may exist, which can be expressed in the form: y = mx + b Equation (4) where m is slope and b is a constant also the y-intercept. To obtain the least squares fit, one has to determine the values of m and b that produce the best straight-line fit for the data. For any straight line there will be a deviation between each value of y as measured experimentally and as determined from the straight line fit for each value of x measured. The least squares fit is that set of values of m and b for which the sum of the squares of

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these deviations is a minimum. Statistical theory states that the appropriate values of m and b that will produce this minimum sum of squares of the deviations are given by the following equations: Equation (5) Equation (6) 3. Percentage Error In cases where the true value of the quantity being measured is known, the accuracy of the experiment can be determined by comparing the experimental result E with the known value K. Normally this will be done by calculating the percentage error of the measurement compared to the given known value, as shown below: Percent error = |E – K| x 100%

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Appendix B (Tables) The tables below are used for reference and comparison purposes with the experimental values the experiment yields. Notice that the helium (He) spectrum has more colors than those tested in the experiment. Because certain colors have lines that are too faint or of similar wavelength and hard to distinguish, they are not easily observed. Since, we are only using helium (He) for calibration purposes, it is unnecessary to sample these colors, and so only the colors in the darkened rows will be observed. APPENDIX TABLE 1: Values of the accepted helium (He) wavelengths

Color in the He Spectrum Wavelength (nm)

Red 706.5

Red 667.8

Yellow 587.6

Green 504.7

Green 501.6

Greenish blue 492.2

Bluish green 471.3

Blue 447.1

Violet 438.8

APPENDIX TABLE 2: Accepted wavelengths of light in Na spectrum

Color in the Na Spectrum Wavelength (nm)

Red 617.2

Yellow 588.4

Yellowish green 567.7

Green 514.1

Green 498.7

D-Lines

Shorter λ Yellow 588.995

Longer λ Yellow 589.592

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Appendix C (The Fabry-Perot Interferometer) The Fabry-Perot interferometer acts as a scanning spectrometer. This system consists of two very flat glass plates coated on the inner surface with a partially transmitting metallic layer. In essence, they act like mirrors that are mounted accurately parallel to each other with a variable spacing. For any fixed distance the interference condition is such that only light of certain corresponding wavelengths will be transmitted. Therefore, this instrument acts as a band-pass frequency filter whose peak transmission is close to unity over a narrow spectral interval. During observation of the transverse Zeeman effect the iris diaphragm is illuminated by the Cd-lamp and acts as such as the light source. The lens L1 and a lens of ƒ = 100 mm, incorporated in the étalon, create a nearly parallel light beam which the Fabry-Perot étalon needs for a proper interference pattern. The étalon contains an interchangeable colour filter which filters out the red cadmium line of 643.8 nm. The lens L2 produces and interference pattern of rings within the plane of the screen with a scale mounted on a slide mount which can laterally be displaced with a prescision of 1/100 of a millimeter. The ring system is observed through L3 and the ring diameters can be measured, for instance, by systematic displacement of the slash representing the “0” of the scale. The readings should be done in a completely darkened room using a flashlight.

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References

(a) Serway, Raymond A. & Beichner, Robert J. Physics for Scientists and Engineers with Modern Physics, 5th Edition, Saunders College Publishing 2000.

(b) PHYWE Publication: University Laboratory Experiments LEP 5.1.10 – Zeeman Effect

LEP 5.1.06 – Fine Structure of One Electron Spectrum (c) “Physics Field Trip”. The Science Man

<http://www.scienceman.com/pgs/archive20_physicsFT.html> (d) Nave, R. “Hydrogen-Like Atoms: Sodium”. HyperPhysics

<http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/sodium.html> (e) Weiss, Michael. “Spin”.

<http://math.ucr.edu/home/baez/spin/spin.html>