Helium Spectrum 3 4 Part 2 example essay topic

Contents 1. Introduction 1.1 Objectives 2. Theory 2.1 Theory of One-Electron Spectrum and Fine Structure of Sodium 2.2 Theory of Zeeman Effort 3. Procedures 3.1 Diffraction Grating Spectrometer 3.2 Fabry-Perot Interferometer 3.3 Part 1: Helium Spectrum 3.4 Part 2: t 1's Order Sodium Spectrum 3.5 Part 3: Yellow D-line in the Sodium Spectrum 3.6 Part 4: Normal Zeeman Effort 4. Experimental Results+ 4.1 Data Table 1: Spectrometer Readings from the He Spectrum 4.2 Calculation Table 1: Grating Constant and its Standard Error 4.3 Calculation Table 2: Data Points for the Calibration Curve 4.4 Data Table 2: 1st Order Spectrometer Readings from the Na Spectrum 4.5 Calculation Table 3: Wavelengths and Percentage Errors using 1st Order Diffraction Grating Equation 4.6 Calculation Table 4: Wavelengths and Percentage Errors using the Calibration Curve 4.7 Data Table 3: Spectrometer Readings for Sodium D-line Spilt 4.8 Calculation Table 5: Sodium D-line Spilt 4.9 Graph 1: Calibration Curve 5.

Discussions 5.1 Possible Causes of Errors and Precaution 5.2 Accuracy of Diffraction Spacing 5.3 Comparison of the Wavelength 5.4 Fine Slitting cannot be observed in the 1st Order Spectrum 5.5 Interference Ring Pattern and the Magnetic Flux Density 6.

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+Experiment Results consist of the log sheets given in the laboratory manual and the graphs drawn based on the findings. Introduction / Theory 1. Introduction It is a well-known fact that the spectral lines of helium (He) can be used to calibrate the diffraction grating spectrometer. The calibrated equipment, in turn, can be used to determine the spectrum of sodium (Na) thus enabling us to find the wavelengths of its spectral lines. Secondly, when a cadmium spectral lamp is subjected to a magnetic field, the red cadmium (Cd) line (λ = 643.8 nm) is split into three components, called the Lorentz triplets. This occurrence is called the "normal Zeeman effect" and can be observed using a Fabry-Perot interferometer. 1.1 Objectives The objective of this experiment is to study the emission spectroscopy and the Zeeman effect. The experiment can be divided into five parts: i) To calibrate the spectrometer using the helium spectrum. ii) To determine the grating constant.) To determine the spectrum of sodium. iv) To determine the fine structure splitting into the sodium spectrum. v) To observe the normal Zeeman effect using the cadmium spectrum. 2. Theory 2.1 Theory of the One-Electron Spectrum and Fine Structure of Sodium 2.1. 1 Discrete or Line Spectrum A hot electrical discharge, according to the electronic structure of that element, emits a spectrum of different colored lights and a limited range of discrete wavelengths in a gas of a single chemical element. This spectrum is known as the discrete spectrum or line spectrum 1. An example of this spectrum is that of helium (He), whose spectrum lines are shown in Figure 1. Color in the He spectrum Wavelength (nm) Red 667.8 Yellow 587.6 Green 501.6 Greenish Blue 492.2 Bluish Green 471.3 Blue 447.1 Figure 1. Wavelengths of light in He spectrum 2.2. 2 Origin of the Sodium (Na) Spectrum The excitation of the Na atoms can be produced by electron impact. When electrons return from the excited level E 1 (high energy state) to the original state E 0 (low energy state), there is an energy difference. This energy difference is released as a photon of frequency f, which can be in the form of x-rays, visible lights etc. 1 The term line spectrum is used because images produced are usually those of a narrow slit, illuminated by the light source. Theory Energy Difference, E 1 - Eo = hf where h = Planck's constant 2 To the first approximation, the electrons of the inner complete shell produce a screening effect on the position-dependent potential V due to the charge on the nucleus. Position-dependant Potential, V (r) = - e 2 Ziff (r) 4 ε 0 r where e is the charge of the electron. The energy levels Enl are similar to those of the hydrogen but with reduced degeneracy of angular momentum. Energy levels, Enl = - m e 4 Z 2 nl 8 h 2 n 2 An approximate formulae for Enl is given below: Enl = - m e 4 8 h 2 (n - μ nl) 2 where μ nl is known as the quantum effect. The quantum effect is dependent on the n and the l. It decreases as the l increases. μ nl values of the Na atom The interaction of the spin of the electron with its orbital moment gives rise to a reduction in the degeneracy of the total angular momentum. j = l + 1/2 ... l - 1/2 where l is the orbital angular momentum of the external momentum. If the interaction term H in the perturbation theory is considered, viz. H = ζ (r) o ^I Then, the approximation formulae for Enl is given by: Enij = Enl + ζ nl 1/2 { j (j + 1) - S (S + 1) - l (l + 1) } And that for the fine structure splitting is given by: Enij = l+1/2 - Enij = l-1/2 = 1/2 (2 l + 1) ζ nl 2 Planck's constant = 6.63 x 10-34 J Theory / Procedures 2.2 Theory of the Zeeman effect The splitting up of the red cadmium (Cd) spectral line at 643.8 nm into three components in the presence of a magnetic field is called the "Normal Zeeman effect". These three lines, also known as the Lorentz triplets, occurs since the Cd atom represents a single system of total spin S = 0. Without the presence of the any magnetic field, there is only one possible D P transition corresponding to a spectral line at 643.8 nm. In the presence of the magnetic field, the associated energy levels spilt into 2 L + 1 components. Radiating transitions between these components are possible, provided that the following selections are met: ML = +1; ML = 0; ML = -1 Although there are 9 permitted transitions, only 3 of them have the same amount of energy and the same wavelength. Therefore, only these 3 lines will be visible to us. The transition of ML = 0 occurs when the spectral line, λ = 643.8 nm. The 1st group where ML = -1 gives the σ -- line whose light is polarized vertically to the magnetic field. The middle group where ML = 0 gives a π -line whose light is polarized parallel to the direction of the field. The last group where ML = +1 gives a σ +-line whose light is again polarized vertically to the magnetic field. This transverse or longitudinal effect Zeeman effect is observed when the magnetic field is transverse or parallel to the emerging light from the etalon respectively. For the radiating electrons, a change in energy E in the presence of a magnetic field is related to the difference in wave numbers /2 of one of the σ -lines with respect to the central line. E = EL, ML - EL-1, ML-1 = hc /2 The change in the energy E is proportional to the magnetic flex density B. E = μ B B where μ B is the Bohr's magnetron. Hence by combining the above two equations, we get the relationship between the difference in the wave number /2 of one of the σ -lines with respect to the central line and the magnetic flux density B: /2 = μ B B hc The above equation shows that /2 is proportional to B and by plotting /2 against B, the Bohr's magnetron μ B can be determined experimentally. 3. Procedures Equipments: Spectrometer / goniometer with vernier Diffraction grating, 590 lines / mm, grating constant d = 1684 nm 3 The Zeeman effect refers to the splitting up of the central spectral lines of the atoms within a magnetic field. Procedures Helium Spectral Lamp Sodium Discharge Lamp Power Supply for Spectral Lamps Lamp Holders for Spectral Lamps Tripod Base 3.1 Diffraction Grating Spectrometer A transmission diffraction grating is a piece of transparent material on which has been ruled a large number of equally spaced slits. The spacing of the slits is called the grating spacing or grating constant and is in the range of 1000 to 2000 nm. If light of wavelength λ falls on to a grating of constant d, it is diffracted. Intensity is maximized when the following condition is satisfied: nλ = dsinθ n, where n 4 is an integer (ie n = 0, 1, 2, ... ) Using the above equation and values from table 1, we can determine the average spacing d between any two consecutive lines of a diffraction grating assuming that the characteristic wavelength of helium is known and the characteristic wavelengths of sodium from the angles θ n where these intensity maximum are observed and measured. 3.2 Diffraction Grating Spectrometer The Fabry-interferometer has a resolution of approximately 300 000, thus it is able of detecting even wavelength changes of approximately 0.002 nm. The interferometer (or etalon) consists of two parallel flat glass plates coated on the inner surface with a partially transmitting metallic layer. Let us consider the two partially transmitting surfaces (1) and (2) separated by a distance t. An incoming ray making an angle θ with the normal to the plates will be split into the rays AB, CD, EF, etc. The path difference between the wave fronts of two adjacent rays will result in the interference rings. The difference in the wave numbers, of two σ -lines can be shown to be: = δ 2 t where δ is the optical path difference between two adjacent rays. Δ is the difference of the squares of the radii of adjacent bright interference rings. In the absence of a magnetic field, a series of bright interference rings corresponding to λ = 643.8 nm is observed. When the field is applied, each ring splits into three rings (i.e. σ +, σ - and π -lines). Since the lights of each spectral light is polarized, whatever the position of the analyzer in the interferometer may be, each of the rings seen without the magnetic field is split into two rings in the presence of a magnetic field. For procedures regarding, 3.3 Part 1: Helium Spectrum 3.4 Part 2: 1st Order Sodium Spectrum 4 n is called the order number. This equation is only valid for integers values of n and for angles θ n up to 90^0. Procedures / Experimental Results 3.5 Part 3: Yellow D-line in the Sodium Spectrum 3.6 Part 4: Normal Zeeman Effort please refer to laboratory manual pages 11 to 14, section 5, Procedures. 4. Experimental Results For 4.1 Data Table 1: Spectrometer Readings from the He Spectrum 4.2 Calculation Table 1: Grating Constant and its Standard Error 4.3 Calculation Table 2: Data Points for the Calibration Curve 4.4 Data Table 2: 1st Order Spectrometer Readings from the Na Spectrum 4.5 Calculation Table 3: Wavelengths and Percentage Errors using 1st Order Diffraction Grating Equation 4.6 Calculation Table 4: Wavelengths and Percentage Errors using the Calibration Curve 4.7 Data Table 3: Spectrometer Readings for Sodium D-line Spilt 4.8 Calculation Table 5: Sodium D-line Spilt 4.9 Graph 1: Calibration Curve Discussions 5. Discussions 5.1 Possible Causes of Errors and Precaution Q. Sources of errors and steps taken to minimize or eliminate them: Firstly, one possible error encountered is the parallax error (random error). This happens while we are taking reading from the vernier scale. If our eye is not placed directly over the reading to be taken and instead, viewed the reading at an angle, there is a slight degree of error. To minimize parallax error, our direction of vision should be placed perpendicular to the scale and directly above reading to be taken. Secondly, the placement of the diffraction grating on the spectrometer can result in systematic error. A slight angle in its position could result in differences between the actual values and the experimental results. Therefore, the diffraction grating should be placed at a right angle to the spectral lamp and as accurate as possible in order to minimize this error. Also, when the reference point is set, a systematic error can occur due to precision of the setting of this point. To minimize these systematic errors, there should be only one person who constantly determines the diffraction angle and another who records readings from the vernier scale. Thirdly, there could be a zero error. There maybe a slight zero error in the vernier scale attached to the spectrometer. In the experiment, we actually try to minimize this error by adjusting the vernier scales and making sure that there is minimum zero error and locking the scales down when recording readings. Next, there should be a better constant between the images and the background. It is best that the background is as dark as possible, preferably black so that the reference point can be aligned accurately with the vertical cross hair. Lastly, one last error that could happen is the backlash error. This happens when one move the telescope backwards to measure an overshot point. There will be a huge error resulting from this measurement, as our vision will be slighted shifted away and distorted. This error can be minimized by shifting the telescope in only one direction and record the measurements. 5.2 Accuracy of Diffraction Spacing Q. Comment on the accuracy of d obtained by calculation and the calibration curve: The value of d obtained by calculation is 1525.27 nm while the standard error of d is 75.29 nm. There is a standard error of 4.9% in the accuracy of the answer obtained from the calculation. On the other hand, the value of d obtained from the calibration curve is 1412 nm while the standard error of d is given by the x-intercept, which is 34 nm. The standard error in is 2.4%. Since the standard error derived from the calibrating curve is smaller as compared to the calculated value, the accuracy of d obtained by the calibration curve is higher (standard error of 2.4%). However, we must bear in mind that the scale of the calibration curve is 1 unit: 0.05 for the y-axis and 4 units: 250 nm for the x-axis. Therefore as the scales are generally larger, the degree of accuracy decreases. So the percentage error of the calibrated curve should be slightly higher than 2.4%. Discussions 5.3 Comparison of the Wavelength Q. Comment on the differences and accuracy of the 1st order Na wavelengths obtained by calculations and from the calibration curve: The calculated wavelengths are relatively accurate as the percentage errors in the calculation of the wavelengths are small. Mostly, it lies lower than 1% except for that of the red image. There is a percentage error of 2.38%, which is relatively higher as compared to the rest of the errors but generally still a low percentage error. The wavelengths determined by the calibration curve are fairly accurate as the percentage errors in the calculation of the wavelengths are small. Mostly, it lies lower than 1% except for that of the red and the second green image. There is a percentage error of 3.29% and 1.52% respectively, which is relatively higher as compared to the rest of the errors but generally still a low percentage error. The accuracy of the wavelengths obtained from the calibration curve is lower than calculation since percentage errors in determination of the wavelengths are relatively higher. This could be because the approximation had to be made while drawing the linear calibration curve joining the point and that the scales of the curve (as mentioned in the question above) is rather large and the points marked are slightly less accurate. 5.4 Fine Slitting cannot be observed in the 1st Order Spectrum Q. Reasons for not being able to observe the fine splitting of the yellow D-line in the 1st order Na spectrum: The reason why we are not able to observe the fine splitting the 1st order is due to the fact that the difference between the diffraction angle of the shorter wavelength and that of the longer wavelength of the yellow D-line is too small. Thus, it is not noticeable by the naked eye. Therefore, sin varies proportionally to the order of spectrum, with the angle, for 1st order spectrum being too small to be seen. The fine splitting of the yellow D-line is best seen with higher orders of the spectrum but unfortunately, in out experiment, we can only managed to observe the spectrum order up to the order of n = 2.5. 5 Interference Ring Pattern and the Magnetic Flux Density Q. Observation and explanation for the dependence of the interference ring pattern in the Zeeman effect on the magnetic flux density. Since the lights of each spectral light is polarized, whatever the position of Discussions / References the analyzer in the interferometer may be, each of the rings seen without the magnetic field is split into two rings in the presence of a magnetic field. As the current is slowly increases from zero with the magnetic flux density increasing slowly at the same time, each ring splits up into three rings. As the current is increased further (magnetic flux increases proportionally to increase in current), each of the triplets become thicker and wider apart from each other. This is because a state of quantum number n breaks up into several sub-states when the atom is in a magnetic field; their energies are slightly more or slightly less than the energy of the state in the absence of the magnetic field. This phenomena leads to a 'splitting' of individual lines into separate lines when atoms radiate in a magnetic field, with the spacing of the lines dependent on the magnitude of the magnetic field.