Protokoll Grundpraktikum II - Optical Spectroscopy 1 Elaboration 1.1 Optical Spectroscopy Student: Hauke Rasch, Martin Borchert Tutor: Madsen Date: 22.October2014 This experiment is about the fundamental principles of optical spectroscopy, as it is a very important method in many researching fields. 1.2 Physical Basics Dispersion The refractive index n, the ratio at which light is refracted when travelling through a transparent medium, for example glass, depends on the light s wavelength. As a result, light of multiple, di erent wavelengths, is split up into its spectral parts. This phenomenon is called dispersion. The dispersion power of a prism is the angle between the two refractions of light at the two ends of the visible spectrum. This dispersion power determines the resolution of the prism and the so called length of its spectrum. Refractive index and dispersion power only depend on the material and not on each other. 1.3 Prism To now simplify things a lot we consider a beam travelling through a prism parallel to one of it s sides. This results in an almost symmetric picture and we can see that and = 2 (1.1) = 2( ) (1.2) 1
Figure 1: Rays in prism http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/imggo/prism.gif 1.4 Resolution Now it is di cult to tell if two rays that have gone through the prism are distinguishable or not. To solve this problem, we now look at the Rayleigh criterion which tells us that this is the case if a di raction-maximum of one line coincides with the minimum of an 8 other line. The intensity of the minimum of this pair of lines then has the value of of 2 the maximum. 1.5 Resolving power The resolution of a prism is finite and depends on the aperture angle of the prism. Here A-A are two wavefronts before entering the prism and B-B are those two after leaving the prism again. For the first maximum, there is no di erence between the wavefronts. For the first minimum (at + ) there has to be a di erence of exactly one wavelength. So for n=1 and with linear approximation we result in 1.6 Grating n ( )=n (1.3) n( + )=n + dn d (1.4) (n + dn d )t nt = (1.5) Now we look at gratings, which can be described as a series of repeated slits in two dimensions. When coherent light shines through this grating, this will cause a di raction and interference pattern on the screen, called Frauenhofer Di raction Pattern. This consist of distinct, sharp maxima that can be located by using the following equation, where d is the distance between the lines of the grating and z is a natural number d sin = z (1.6) 2
1.7 Resolution of the grating One of the big advantages of this pattern is that the all the main maxima are very sharp because higher order minor maxima are decreasing quickly in intensity. The first minima for each main maximum can be found by looking at this equation, where N is the total number of contributing slits. With the Rayleigh criterion from above we find that dsin min =(z + 1 N ) (1.7) = zn (1.8) = zn (1.9) This means that the resolution increases with the number of grid gaps. 1.8 Balmer Series and Rydberg In 1885 Balmer discovered, that one can calculate the frequency of the Hydrogen s spectral lines with the following equation. v = 1 = R( 1 1 2 2 ),n=3, 4, 5,... (1.10) n2 R is called Rydberg-constant and is defined as follows. R = 2 2 m e e 4 h 3 c This equation becomes the base for the nuclear model of Bohr. (1.11) 3
2 Measurement Protocol Time and Date: 22. October 15:48-18:35 Tutor: Madsen Students: Hauke Rasch, Martin Borchert Errors, or rather uncertainties when measuring All angle measurements: ±0.02 2.1 Exercise number 1 The Spectrometer was set up. This will be explained further in the evaluation. 2.2 Exercise number 2 Zeroth order maxima: The zeroth order is located at an angle of about 178.08. First order and second order maxima right side right in degrees left in degrees in degrees Subjective colour impression in nm 192, 11 164, 04 14, 035 bright violet 404, 7nm 192, 25 163, 92 14, 165 weak violet 407, 8nm 193, 21 162, 92 15, 145 blue / violet 435, 8nm 195, 24 160, 93 17, 155 turquoise / green 491, 6nm 197, 18 158, 96 19, 110 strong green 546, 1nm 198, 31 157, 86 20, 225 first yellow 577, 0nm 198, 40 157, 76 20, 230 second yellow 579, 1nm 200, 02 156, 11 21, 955 (orange) red 623, 4nm 202, 55 153, 60 24, 475 weak red 690, 7nm 207, 00 149, 05 28, 985 bright violet (2nd) 404, 7nm 207, 35 148, 80 29, 275 weak violet (2nd) 407, 8nm 209, 58 146, 55 31, 515 blue / violet (2nd) 435, 8nm 214, 21 141, 92 36, 145 turquoise / green (2nd) 491, 6nm 218, 97 137, 14 40, 915 very strong green (2nd) 546, 1nm 221, 85 134, 28 43, 785 first yellow (2nd) 577, 0nm 222, 05 134, 07 43, 99 second yellow (2nd) 579, 1nm 226, 45 129, 67 48, 390 (orange) red (2nd) 623, 4nm 4
2.3 Unknown Lamp Zeroth order maximum: The zeroth order is now located at the angle of about 178, 04. First order maxima: Subjective colour impression right in degrees violet 193, 39 blue 194, 34 turqouise 194, 77 green 195, 79 weak red 199, 49 normal red 200, 32 strong red 200, 74 2.4 resolving power Measurement were done on the right side and for first order maxima. Totally closed slit at: 5100 10 6 m Open slit at: 4610 10 6 m Slit opening distance: l = 490 ± 10 10 6 m 5
Figure 2: Basic setting 3 Evaluation First we adjusted the spectrometer. Therefore the lamp was switched on and right in front of it we placed a thin, variable slit and in front of that a collimator lens. To retrieve good image quality at first a mirror was placed on top of the prism table, where the grating is later going to be. After that all the devices were aligned and positioned properly in a row until the light that had shone through the slit and the collimator and had been reflected by the mirror would perfectly pass back trough the collimator and back into the slit, resulting in a sharp line in the plane of the slit. By doing this, one makes sure that the slit is placed correctly at the focal point. The mirror was carefully switched for the grating. This was a bit tedious as there is no way to properly fix the grating frame onto the prism stand, as the holder can turn freely. Thereafter behind the grating we placed an ocular and an objective. The objective lens was moved to its correct position to create a sharp image in the image plane. An important factor in adjusting all these devices was that they all had to be on exactly the same height and that, after it was once set, it had not to be moved ever again. This took some time and e ort to do. 6
Figure 3: Spectral lines through the ocular 3.1 Finding the grating constant d To calculate the grating constant, which was unknown, we measured the angle of the zeroth order maximum. This maximum was at 0 = 178.08 and just as sharp as the other spectral lines. Then the goniometer was moved towards the right side until the lines of the spectrum appeared inside the ocular. Then we carefully moved the goniometer until the crosshairs inside the ocular lined up with the spectral lines. Then the angle was measured and written down in the measurement protocol, next to it we made a note what colour the line looks like and the intensity relative to its neighbouring lines. Also we looked up what wavelength of each line by comparing it to the known Spectrum of the Hg-Lamp from the script. This wavelength was also written down in the protocol. The same procedure was repeated on the left side. The angle of di raction was calculated as half the di erence between left and right. The zeroth order was only measured to ensure the symmetry of the spectrum, which was found to be very symmetric with an error of ±0.02. = right left (3.1) 2 7
Figure 4: Spectral lines through the ocular We estimate the uncertainty of right and left to be left/right = ±0.02. The error of this angle is calculated as one half of the root of the sum of the squares of the errors of the measurements of left and right. p ( right ) 2 +( left ) 2 = (3.2) p 2 (0.02 ) = 2 +(0.02 ) 2 (3.3) p 2 8 = 2 0.01 = p 2 0.01 =0.014 (3.4) The grating constant is calculated by. d = z sin (3.5) The error of the grating constant only depends on the error of the angle measurement, because both the wavelengths, which are literature values, and the number of order do not have errors. Therefore the error is retrieved as follows. d = z sin 2 cos = d tan (3.6) 8
The following two charts depict the measurements for the first and second order maximum. in degrees in nm d in nm d in nm 14,035 404,7 1668,8 1,7 14,165 407,8 1666,4 1,6 15,145 435,8 1668,1 1,5 17,155 491,6 1666,7 1,4 19,110 546,1 1668,1 1,2 20,225 577,0 1669,0 1,2 20,230 579,1 1674,7 1,2 21,955 623,4 1667,4 1,1 24,475 690,7 1667,2 1,0 Table 1: First order maximum in degrees in nm d in nm d in nm 28,985 404,7 1670,3 1,6 29,275 407,8 1667,9 1,6 31,515 435,8 1667,4 1,5 36,145 491,6 1666,9 1,3 40,915 546,1 1667,6 1,1 43,785 577,0 1667,7 1,1 43,990 579,1 1667,6 1,1 48,390 623,4 1667,6 1,0 Table 2: Second order maximum The average of all the grating constants calculates to d = 1668.16nm. Error: From the gaussian error propagation we conclude an error of d = 0.31nm. 9
3.2 An unknown lamp Now the Hg lamp was replaced by an unknown lamp with a di erent spectrum. Again the angles under which the spectral lines could be located were measured and written down. Therefore the zeroth order maximum was measured again and found to be at 178.04. This time measurements were only done on the right side and only for maxima of the first order, as these were a lot better visible than the ones on the left side. As only one value for each wavelength was determined, the error doubles to ±0.028. The average value for the grating constant d = 1668.16nm from above was used to calculate the wavelength of the light that was observed under di erent angles. The formula to calculate deducts from (3.5). d = z sin = d sin z (3.7) The error is derived with the gaussian error propagation from the formula above. r = ( sin d) z 2 +( d cos ) z 2 (3.8) The calculated wavelengths were compared to di erent spectra of di erent lamps. Cadmium were found to have the very best fitting spectrum. Below both are compared in a table. right in in in nm in nm for Cd-lamp in nm distance to in nm 193,39 15,35 441,6 0,8 441,5 +0,1 194,34 16,30 468,2 0,8 467,8 +0,4 194,77 16,73 480,2 0,8 480,0 +0,2 195,79 17,75 508,6 0,8 508,6 ±0, 0 199,49 21,45 610,0 0,8 609,9 +0,1 200,32 22,28 632,4 0,8 632,5 0,1 200,74 22,70 643,8 0,8 643,8 ±0, 0 Table 3: Unknown lamp - angles of spectral lines All the values for the wavelengths of the spectral lines of a Cadmium lamp fall into the single error interval of these measured values. 10
Figure 5: Spectral lines of Helium, Cadmium, Zinc 3.3 Resolving Power The mercury lamp was put back into the setup and a slit was placed right in front of the grating. The two yellow lines from the Hg spectrum were observed again through the lens at 198, 31 and 198, 40. Those two lines have the wavelengths of 1 = 577.0nm and 2 = 579.1nm, so they are = 2.1nm apart from each other. The average between 1 and 2 is = 578.1nm. The slit was closed all the way and very slowly opened up until the two yellow lines were barely distinguishable. At this point the slit was opened l = 490 ± 10 10 6 m. Given the previously calculated grating constant we find out that zn = l d z = 490000nm 1 = 293, 73 = 294 (3.9) 1668, 16nm So we are looking at 294 lines of the grating.? =zn (3.10) 578.1nm? =294 2.1nm (3.11) 289 294 (3.12) 11
From gaussian error propagation the uncertainty for N and hence for zn can be derived as follows, because z does not have an uncertainty. r (zn) = ( 1 d l)2 +( l d 2 d) 2 (3.13) s 1 (zn) = ( 1668, 16nm 10, 490, 000nm 000nm)2 +( (1668, 16nm) 2 0.31nm)2 (3.14) (zn) =5, 99 = 6 (3.15) Once again within the first error interval the values are equal. Further explanation for this problem and for the second order can be found further down under Summary and Discussion. Figure 6: Micrometer screw with closed slit 12
3.4 Qualitative discussion of the prism The grating was taken out of the apparatus and replaced by a prism. The observations were written down and can be found in the Summary and Discussion section below. Figure 7: Dispersion of white light when passing through a prism www.heasarc.nasa.gov/docs/xte/learning_center/universe/gifs/prism3.gif 13
4 Summary and Discussion 4.1 The Grating Constant With our measurements and the help of known spectral lines of a mercury lamp the grating constant of an unknown grating was calculated. The average of the grating constant was determined to be 1668.2 ± 0.4nm 4.2 The Unknown Cadmium Lamp In the next experiment the spectrum of an unknown lamp was measured and, together with the knowledge about the grating constant from above and the known spectral lines of several di erent lamps, the lamp was found to almost certainly be a helium lamp. 4.3 The Resolving power The resolving power of this setup was measured by closing a slit in front of the grating and seeing when the two characteristic yellow lines of the mercury lamp would merge into one undistinguishable line. The theoretical value and the experimental value 289 and 294 are matching each other. Still one of the biggest problems was that it is not possible to put the slit exactly in front of the grating because of the holders for both devices. So if the slit is a bit further away from the grating the area on the grating that light can shine on and hence the number of grating lines increases, thus giving a result di erent from the expected one. Though this problem is of a minor sort and the result for this task is quite pleasing. 4.4 The Di erent Prism After that the grating was replaced by a prism to qualitatively find the di erences between those two. These di erences were that the prism did not show any interference patterns (higher orders) and it s dispersion is smaller than the grating s one. The colour spectrum was flipped between both. With the prism larger wavelengths are less refracted than smaller wavelengths. (Compare Figure 7) 14