Experiment :O-9 Determination of the wavelengths of the Sodium doublet lines and the measurement of the separation between the D 1 and D 2 lines using a Michelson interferometer. Submitted by Muhammed Mehedi Hassan Group A ;Batch-09 Second Year, Roll SH 236 Student of Physics Department, Uinversity of Dhaka. Date of experiment September 09, 2011. Date of submission September 27, 2011.
Experiment :O -9 Determination of the wavelengths of the Sodium doublet lines and the measurement of the separation between the D 1 and D 2 lines using a Michelson interferometer. Theory : A schematic view of a Michelson interferometer is depicted in Figure 1. Light from the source enters the interferometer and encounters a beam splitter. A portion of the light continues on to reflect from mirror M 2, this reflected ray from the beam splitter fall into the detector. The other portion of light is reflected by the beam splitter towards the movable mirrorm 1, this mirror can be adjusted to increase or decrease the optical path difference between the paths taken by the two fractions of the initial light. Having reflected from mirror M 1 this light passes back through the beam splitter to the observer. Figure 1: Schematic of a Michelson interferometer. The light path is shown by the arrowed lines. Before measurements can be taken, it is necessary to perform a calibration of the apparatus. Firstly, an adjustment for circular fringes is made, this brings a set of interference fringes into the field of view. A diagrammatic view of the interference fringes obtained is shown in Figure 2. Figure 2: Interference rings from the Michelson interferometer. We know, if the distance between n fringes is x then this two line determines the wavelength of sodium light, with the following equation: nλ = 2x cos θ (1) Here the angle θ is very small and thus we can rewrite this equation, λ = 2x n (2) 2
As was mentioned before, the use of a lever arm to move mirror M 1 necessitates the determination of a correction factor to ascertain the actual change in the optical path length caused by the movement of the micrometer screw gauge. Since every fringe passing indicates a change in the optical path length, the correction factor f can be found using the general error propagation equation. The correction factor found to be: f = 1 10 = 0.1 (no units) Now equation (2) can be rewrite with this correction factor and it becomes: λ = 2x nf (3) It can be found that for light of two very similar wavelengths (as we have here) the difference between the wavelengths is: λ = λ2 2d s f (4) Where, λ is the difference in wavelength between the doublet lines. λ is the mean wavelength of the doublet The micrometer movements d s will be so as to move the mirror between two points of zero optical path difference. This is signied by the disappearance of the interference fringes and the appearance of a uniform colour. By measuring the distance between successive states of zero optical path difference, a value for the wavelength difference between the two sodium doublet lines can be found and, since the mean wavelength is known, the wavelengths of them can be deduced. Apparatus : 1. A sodium lamp 2. A optical banch 3. A Michelson s interferometer 4. An eye-piece. Least count : Least count= pitch circularscaledivisions Therefore, L.C.= 1 50 2 mm. =0.01 mm 3
Table -1:Determination of the distance x for shifting of 20 fringes : No. Main Circular Total Distance Mesn of scale scale fringes reading reading reading x x σ x mm mm mm mm mm mm 10 35 10.35 n 10 35 10.35 10 35 10.35 10 41 10.41 0.06 0.062 σ x1 =0.00200 n+20 10 42 10.42 0.07 0.062 σ x2 =0.00200 10 41 10.41 0.06 0.065 σ x3 =0.00245 10 47 10.47 0.06 n+40 10 48 10.48 0.06 10 48 10.48 0.07 10 53 10.53 0.06 n+60 10 54 10.54 0.06 10 55 10.55 0.07 10 60 10.60 0.07 n+80 10 60 10.60 0.06 10 61 10.61 0.06 10 66 10.66 0.06 n+100 10 66 10.66 0.06 10 67 10.67 0.07 Table -2:Determination of dissonance d s : Obs. L.S.R. C.S.R. Total Difference Mean d d no. mm mm mm mm 1. 10 89 10.89 0.03 2. 10 86 10.86 3. 10 83 10.83 0.04 0.033±.0033 4. 10 79 10.79 5. 10 76 10.76 0.03 6. 10 73 10.73 4
Calculation : Here, x 1 = 0.062 cm; σ x1 =0.0020 cm x 2 = 0.062 cm; σ x1 =0.0020 cm x 3 = 0.065 cm; σ x1 =0.00245 cm Therefore: x= x 1 + x 2 σx 1 + x 3 σx 2 σx 3 1 σx 2 + 1 σ 2 + 1 = 15500+15500+10995.42 1 x σ 2 2 x 3 666597.25 mm=0.062 mm From equation (3) wavelength of sodium light is: λ = 2 x n f = 2 0.062 20 10 = 6.2 10 4 = 620 nm Separation λ: From eequation (4): λ = λ2 2d sf = (6.2 10 5 ) 2 2 0.333 10 = 5.99 101 7 mm =0.599 nm=5.99 A Error for λ: 1 = 1 + 1 + 1 =666597.25 mm 2 σx 2 σx 2 1 σx 2 2 σx 2 3 σ x =1.225 10 3 mm σ 2 λ= ( λ x )2.σ 2 x=( 2 20 10 )2 (0.001225) 2 = 1.26500 10 9 mm 2 σ λ =3.6225 10 5 mm=36.22 nm Error for separation: σ 2 λ=( λ d ) (σ d) 2 = 7.296 10 16 mm 2 σ λ = 2.27011 10 8 mm=0.27 A 5
Result : The wavelength of sodium light=620 ± 36.22 nm. And the separation between the two lines=0.599±0.027 nm The actual wavelength of sodium light=589.0 nm And the separation between the two lines=0.6 nm The percentage of error for wavelength of sodium light= 620.0 589.0 589.0 100%=5.26% The percentage of error for separation between the two lines= 0.600 0.599 0.600 100%=0.17% Discussion : Human error in counting the fringes adds to errors. This could be improved by having a program to count the fringes, thus precluding the possibility of a miscount in numbers. It has been explained later. The magnification of the screen could have been greater, to ensure that no fringes were skipped in the counting off. Another reason of the error is the correction factor, as we have taken from a previous experiment which was not performed recently, so there lies some change in the value of the correction factor. In the measurement of the dissonance (d) between the two consecutive appearence and disappearence do not differ largely. If we took more sample data that will reduce the error in measuring d. The main problem here was backlash from the micrometer screw gauge, another problem encountered was the possibility of over or under counting the amount of fringes that had passed due to a small movement of the micrometer screw gauge causing a large number of fringes to pass. Finally it was highly possible to knock the micrometer screw gauge when releasing ones grip on it, causing another error. Unfortunately it is very hard, if not impossible, to put a number on these human errors and as such whilst they are definitely there they are hard to take into account when determining a final answer. Correcting these errors The best method, it would seem, to removing these errors would be to have a computer connected to a camera and motor assembly take the readings. Using a motor would prevent backlash and other problems with the micrometer screw gauge, while having a computer count the fringes would reduce counting errors. This would help remove the element of human element present and make any errors occurring easier to place a numerical value on. In the measurement of the dissonance, when the fringes were totally invisible, sometimes we were missing due to its narrow span. Again these are mainly human errors and are very hard, if not impossible, to put a numerical value on. Correcting these errors A computer with a motor and camera assembly would again do much to solve these problems, both for the reasons specified above. 6