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by: Dr. Simeon Wiza


Dr. Simeon Wiza
GPA 3.96


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This 10 page Class Notes was uploaded by Dr. Simeon Wiza on Wednesday September 9, 2015. The Class Notes belongs to PHYS 331 at University of Washington taught by Staff in Fall. Since its upload, it has received 22 views. For similar materials see /class/192458/phys-331-university-of-washington in Physics 2 at University of Washington.




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Date Created: 09/09/15
Physics 331A Experiment 2 CONCAVE DIFFRACTION GRATING SPECTROGRAPH Revised May 16 2006 Diffraction gratings are widely used for high resolution spectral studies The grating available in the lab is a holographicallyproduced concave grating of 600 linesmm and radius R 1075 mm It is placed in a Rowland mounting In such a mount the grating is constrained to be on a circle of diameter equal to the radius of curvature of the grating the Rowland circle on which the entrance slit the slit through which light is incident on the grating is also located If the entrance slit and the grating are on the circle the slit image will be in focus on the same circle Concave gratings are often used in the ultraviolet portion of the spectrum where their focusing properties eliminate the need for UVtransmitting lenses In this experiment strong lines from the visible part of the spectrum of the elements mercury and cadmium are used to calibrate the spectrograph The spectrograph is then used to measure the wavelengths of the lines in the visible spectrum of hydrogen Balmer series from which a value of the Rydberg constant for hydrogen can be obtained The lamp used also contains the isotope of hydogen deuterium so the spectrum of deuterium can be obtained as well for this part we use a CCD array detector which allows a more accurate measurement to be made of the separation betwen similar lines of the two isotopes and also allows the experimenter to make a quantitative measurement of the resolving power of the spectrograph REFERENCES H Hecht Optics 4th ed pp 4767481 general properties F Thorne Anne PSpect rophysics 2nd ed Chap 6 pp 1447170 types of diffraction gratings and mounts focusing property of concave gratings i 3 Davis Sumner Pi Di mction Grating Spectrographs practical information on the alignment and aux iliary optics for diffraction gratings tb Tipler Modem Physics pp 144151 reduced mass correction APPARATUS The apparatus is pictured in the Fig l with the Rowland Circle shown as a dashed line The radius S of the grating is 1075 mm i 2 mm the width of the grating is 25 mm and the grating constant is 600 linesmm error unknown The grating is blazed to give maximum intensity in the rst order The grating mounting has been carefully adjusted and should not be disturbed A movable carriage holds a viewing microscope and CCD array which can be focused on the slit image A set of crosshairs in the microscope allows for precise location of the spectral line under observation The distance x from the slit to the slit image can be measured on a wooden meter stick associated vernier to a precision of about 01 mm For a Rowland mounting with the angle of re ection ice the angle between the grating normal and the axis of the detector 9 0 the grating formula becomes dsinl9 51 m where d is the linetoline separation on the grating 19 is the angle of incidence of the light on the grating 95 is the slit to image distance S is the gratingtoimage distance m is the order of the spectrum m l 2 etc and is the wavelength of the spectral line CALIBRATION OF THE SPECTROGRAPH The wavelength of a spectral line can be determined from a measurement of x and use of the grating equation but this method has signi cant limitations The grating equation is only as precise as the values of d and S and it depends on positioning the entrance slit exactly above the zero of the meter stick scale a result difficult to achieve in practice Physics 331A Experiment 2 Contave y y Rowland Urde Figure 1 The concave grating spectrograph The carriages With the grating and detector microscope or CCD array move perpendicular to each other along tracks so that the distance between them S stays constant even though the angle 9 changes The image of the slit is focused by the grating Which is slightly concave on the detector Because S stays constant the spectrograph stays in focus over the range of angles 9 The grating detector and entrance slit lie on an imaginary circle of xed diameter called the Rowland circle Physics 331A Experiment 2 To avoid these limitations the spectrograph is calibrated by measuring 95 for a number of spectral lines the wavelengths of which are known precisely An empirical relation is then established from which the wavelength of an unknown line as a function of x can be obtained more precisely than it can from the grating equation The wavelengths for the spectral lines of mercury and cadmium that may be used to calibrate the spectrograph are listed below A Element Color You do not need to measure all of these lines but pick at least 5 widelyspaced lines to give a good leverarm to your calibration graph The calibration obtained from the measurements of these lines is to be used for a l t gt UUIJl I PROCEDURE Position the large carriage supporting the mercurycadmium lamp assembly so that the small piece of white tape on the left edge of the carriage is above the 570 mm mark on the metal rail and c amp the carriage securely to the rail with the silver clamping knob If the hydrogen lamp is in the way slide it back and clamp it to the rail before positioning the mercurycadmium lamp The hydrogen lamp assembly is delicate 7 please handle it carefully Position the smaller carriage directly beneath the mercurycadmium lamp housing so that the small piece of yellow tape on the near edge of the carriage is above the 179 mm mark on the small metal rail and clamp this carriage in position Turn on the power supply for the mercurycadmium lamp Check that the lens is positioned so that the small piece of white tape on the left edge of its carriage is above the 835 mm mark on the metal rail With the lamp and lens in these positions the lamp image is focused on the slit with 11 magni cation to see the image hold a white card next to the slit Adjust the slit width using the small lever marked with red tape to approximately 02 mm The width scale is marked on the slit housing For x m 35 cm the green line should be clearly visible Before looking through the microscope make sure that its base is rmly seated on the platform The springloaded screws on the adjacent rails insure that the microscope does not shift on the platform If the observed line is not in focus adjust the focus with the large focusing knob on the right side of the microscope assembly Maximize the brightness of the line by adjusting the transverse position of the lens using the knob marked with yellow tape Also move the small carriage beneath the mercury cadmium lamp housing back and forth to maximize the brightness of the line and again clamp the carriage in position The slit width can now be decreased to allow for more precise location of the center of the line 1 Make ve or more independent measurements of the position of the green line Move the carriage with the microscope well away from the line then reset it so the crosshairs are centered on the line From these measurements determine the precision with which you can locate a line Each student in the group should make this set of measurements Physics 331A Experiment 2 2 Compute the positions of the calibration lines in the rst order e m 1 from the grating equation These computed positions need not be especially precise but they will help in identifying the correct calibration lines 3 Carefully record the positions of the calibration lines in rst order Plot each point carefully on the graph paper provided after you measure a line Plot as in cm on the vertical and in nm on the horizontal Choose the scales for ac and so that the data nearly ll the page Visually verify that a spectral line of the right color exists at that value of 95 Use a straight edge to draw a line through the data points If you have correctly identi ed each line and measured its position carefully the plotted points will lie on a straight line as closely as you can plot them If not remeasure the lines that are 0 When you have nished recording the calibration lines look at several of them green amp yellow in the second orderl Notice how much dimmer these lines are than their rst order counterpartsl This is the result of the grating being blazed so that maximum intensity is re ected in the rst orderl After looking at the second order lines turn off the mercurycadmium lampl Later when you write your report also do the following 4 Use EXCEL or another curve tting program such as KaleidaGraph or Mathematica to t a least squares straight line to the calibration data Obtain the slope s the standard deviation in the slope as the intercept 950 the standard deviation in the intercept 710 and the standard deviation in an individual measurement of as oil Comment on the value of the intercept you obtained ls it reasonable Why might the intercept be nonzero 5 From the slope and the radius R calculate the lines per mm in the grating and the standard deviation in this number Compare this to the nominal value of 600 mm l and comment on how any difference might arise BALMER SERIES AND THE RYDBERG CONSTANTS The visible spectrum of hydrogen consists of a series of lines that result from transitions terminating on the energy level with principal quantum number n 2 and originating from higherlying energy states for which n 3 4 5 etc The series of lines is named after Johann Balmer who in 1855 published a numerical relation accurately describing their wavelengths Starting with the longest red wavelength the lines are labeled Ha Hg Hq etc The Balmer series wavelengths for hydrogen are described by the Balmer formula 1 1 1 X ag 27 n345m 1 The constant is called the Rydberg constant for hydrogen A similar expression applies for deuterium but the Rydberg constant for deuterium BB is slightly larger than that for hydrogen The difference arises from the effect of the greater mass of the deuterium nucleus A larger nuclear mass results in a larger value for he the electron reduced mass The reduced mass of the electron arises from the fact that the twobody problem of an electron and a nucleus can be recast into an effective onebody problem involving the reduced mass of an electron only The formula for the reduced mass is e 2 Mg 1 meM where in3 is the mass of an electron and M is the mass of the nucleus See the reference by Tipler and the material on the class website for the derivation of this formula The Bohr and Schrodinger theories of the hydrogen spectra give a formula for the Rydberg constant 4 R K 3 7 Seghgc 4 Physics 331A Experiment 2 where e is the charge of an electron 50 is the permittivity of free space 1 is Planck s constant and c is the speed of light The Rydberg constant for an in nitely heavy nucleus R00 which is the same as Eq 3 with be me is one of the best known constants in physics RDO 10 973 731568 54983 m 1 CODATA recommended values of the fundamental constants 1998 Note that this is larger by a small amount than the value of the Rydberg constant for hydrogen In the next exercise you will take data in order to determine RH Move the large carriage supporting the mercury cadmium lamp toward the lens carriage and clamp it a cm or so from the lens carriage Move the small carriage and mercury cadmium lamp housing toward the wall out of the light path and clamp it in position Move the carriage supporting the hydrogen deuterium lamp assembly forward until the small piece of white tape on the left edge of the carriage is above the 515 mm mark on the metal rail and clamp the carriage with the silver clamping knob With the lamp and lens in these positions the lamp image is focused on the slit with 11 magni cation to see the image hold a white card next to the slit You may need to make some adjustments of the lens in order to optimize the position and the focus of the image of the hydrogen deuterium lamp The hydrogendeuterium lamp gets very hot after being on for a period of time and as its temperature increases the intensity of the emitted light decreases To help keep the lamp cooler and the intensity higher a small fan is positioned at the base of the black tubing which protects the lamp Turn on the power supply for the fan and then turn on the power supply for the lamp For x m 42 cm the red Balmer a line should be visible If the line is not uniformly bright it may be necessary to adjust the transverse position of the lens andor the longitudinal position of of the hydro gen deuterium lamp The hydrogendeuterium lamp draws 20 milliamperes mA of current at 5000 volts potential difference across the tube 20 mA can be a lethal current so it is important to handle the hydrogendeuterium lamp assembly carefully and touch it only at grounded surfaces To change the position of the lamp slightly loosen the knob marked with blue tape and rotate the lamp holder by grasping the short horizontal rod marked with orange tape Do not touch the black tubing protecting the lamp or the white caps holding the tube in place When you have achieved maximum brightness by adjusting the lens and lamp positions clamp the lamp in position by tightening the knob marked with b ue tape Once you have found the Balmer oz line reduce the slit width in order to see the separation between the hydrogen and deuterium lines Which line corresponds with which element Hint study Eqs 1 through 2 When you have gured out which lines are the hydrogen lines focus the crosshairs of the microscope on those lines Measure the position of the three visible Balmer lines 1 and y for hydrogen in rst order The other lines in the Balmer series fall outside the visible range Note there are a number of lines in the spectrum of the lamp that come from impurities rather than from hydrogen or deuterium These tend to be notably dimmer than the hydrogen lines but you can use your calibration from the mercurycadmium lamp to make sure you are observing the correct lines 6 Determine the value of RH from measurements of the Balme39r lines of hydrogen Extract the Rydberg constant for hydrogen RH from the parameters of a straightline t done with EXCEL or other program to an appropriately chosen plot of your data Hint The Balmer formula can be reduced to a linear equation by substituting the proper functions of and 71 See Eq Calculate an uncertainty in your value of RH and compare your result with the accepted value of ROC adjusted for the reduced mass of hydrogen Physics 331A Experiment 2 THE ISOTOPE EFFECT amp THE RESOLVING POWER OF THE SPECTROGRAPH The ability of the spectrograph system to resolve small differences in the spectral lines of a source depends on many factors the quality of the optical components and their alignment the width of the entrance slit the quality and properties of the grating the resolution of the detector system whether it is someone s eye or an electric photodetector and the inherent properties of the light source The resolving power R of a diffraction spectrograph is described by the formula R 4 mm f l where Akmin is the wavelength difference between two spectral lines that are just resolved ilel seen as separate lines The contribution of the grating itself to the resolving power is given by theory for a perfectly made grating to be Rgrating quotIN a 5 where m is the order number and N is the total number of lines in the grating that are illuminated by the light source In practice the resolving power will always be less than this theoretical va uel In this next exercise you will use a CCD ChargeCoupled Device array to measure the separation between the Balmer lines of hydrogen and deuteriuml The fact that there is a separation is called the isotope effect You will also estimate the resolving power of the spectrograph from the width speci cally the fullwidth at halfmaximum of these spectral lines The CCD array has 1024 equally spaced sensing elements pixels in the space of 799 mm so it is capable of resolving very closely spaced lines The array electronics reads ou the array periodically according to an internal clock circuit The output of the array circuit is fed to a homemade ampli er that allows a DC offset to be applied to the signall The ampli er output is sent to a digital oscilloscope A second output from the array circuit is used to trigger the oscilloscope to produce a stable tracel We will use the features of the oscilloscope to reduce noise in the CCD signal Before switching to the array you should make sure you can see the two lines with your eye Position the microscope carriage to observe Ha linel Adjust the slit width until you see two very closely spaced red lines these are the Ha and DEX lines The CCD array will allow you to make a quantitative measurement of their separation Carefully lift the microscope assembly and set it aside taking care not to move the microscopedetector carriagel Loosen the silver clamping knob on the black post holder and swing the CCD array board into position so that the plane of the board is perpendicular to the light path Clamp the array in position The sensing elements of the array are beneath a glass cover on a small chip near the lower center of the circuit cardl Turn on the power to the oscilloscope and the power supply connected to the CCD card and ampli er boxl Grab a ashlight and turn off the room lights the CCD is very sensitive and you ll need to work in the dark for this part Set the scope sweep speed to about 10 milliseconds ms per division and the vertical gain to about 05 volt per divisionl Check that the trigger menu shows the scope on external EXT trigger with a trigger level between 200 and 250 mV and that the Acquire menu is set to Sample If you do not see a bump in the trace that moves as you slide the carriage a small amount open the slit up completely and hold a white card in front of the CCD card and look for the hydrogen lines If the room is dark you should be able to see the red oz and turquoise linesl Use the card to help you see where to move the carriage so that the red line crosses the detector window You should now be able to see a large bump in the signal trace on the scope similar to the signal shown in Fig 2a If you can t nd the signal ask for help Physics 331A Experiment 2 1 11 Acquisition 39 39 1 I 39 39 11 Mode Peak Detect lt 250MSKS l 39500mv 40b Ext39 x 23904mvi Z I g I I I Q 2221 4000 9a g g j 500mv M Al Ext quotL 204mV 39 Mode HorizontaI Reset Acquisitions 259 Avera 8 Resolution Horizontal Autoset Sample Rate 250MSs 3800 9s 9 Normal Delay Figure 2 Scope traces showing the CCD array output a The Balmer oz lines with the entrance slit too wide to resolve the difference between HO and D0 Note the m 25 V readout cycle start spike directly below the trigger marker T b The oz lines resolved into the HO and DO components Note that the carriage has been moved so that the lines are near the start cycle spike and the horizontal and vertical scales have been greatly increased so that the steps corresponding to individual pixels are visible Once you can see the signal of the oz lines reduce the slit width enough to split the large bump into two smaller bumps Then loosen the silver knob on the side of the post holder that supports the CCD card assembly and while watching the scope slowly rotate the card assembly back and forth to nd the position which makes the peaks the highest and narrowest possible This optimizes the position of the CCD detector so that it is at the focus point of the grating Next slide the carriage slowly along the track to position the peaks near the beginning of the CCD readout cycle which is marked by the short spike or blip of about 2 volts amplitude See Fig 2a The vertical position of the trace on the scope is determined by two knobs the vertical Position knob on the sc0pe panel sets the position of zero volts as denoted by the channel marker on the left side of the display and the Offset knob on the amplifier box which applies a DC offset to the CCD signal It is helpful to position the channel marker about two large divisions below the centerline of the display and then adjust the Offset knob to park the trace at this spot This is shown in Fig 2a note the marker for channel 1 at the left Increase the sweep speed to about 400 uS div in order to zoom in on the peaks Then make successive small adjustments of the CCD card position carriage position slit width and oscilloscope controls in order to make the peaks on the scope as narrow and as distinct as possible You should be able to obtain an image similar to that shown in Fig 2b Note the significant change in both the vertical sensitivity CH 1 and horizontal sweep speed M between Figs 2a and 2b When you have zoomed in on the peaks the trace will be quite jumpy because of electrical noise You can quiet this noise by using the averaging feature of the scope Press the MENU button in the ACQUIRE column then press the Average button to the right of the scope screen next to the Average selection An average of 32 sweeps will make for a steady display To change the number of sweeps in the average rotate the knob with the G9 on it at the upper right of the screen Trace averaging was done in Fig 2b At this point the HO and DO lines should be clearly resolved The readout from individual pixels should now be visible on the scope each pixel shows up as a flat section with a faint spike at the edge A very nice hard copy of the CCD readout can be obtained with the Tektronix T DS 3012 digital sc0pe If there are waveforms displayed other than Channel 1 yellow you can remove them from the display by pushing the appropriately colored button white for REF red for MATH and then pushing the OFF Physics 331A Experiment 2 button To print the scope display press the button at the lower left of the screen with the printer icon on it retrieve your print from the printer below the scope By counting pixels the stairsteps in Fig 2b you can measure the separation between the two peaks and estimate the fullwidth at half maximum of the peaks But to relate the pixel counts to the change in wavelength you need a conversion factor which you can determine by using the CCD array and the alpha lines as follows Set the sweep speed back to 10 msdiv so that you can see an entire readout cycle of the CCD array Move the carriage to the left so that the very top of the largest peak is just at the left edge of the CCD readout Note the at reading call it 95177 on the meter stick Now move the carriage to the right so that the very top of the same peak is just at the right edge of the CCD readout and again note the at reading 952 on the meter stick Hint if you understand how this system works you can get a more accurate result by moving the horizontal position marker or trigger point77 to the middle of the screen and then increasing the sweep speed sufficient to see individual pixels Then at one extreme the peak will be on the right side of the cyclestart spike and at the other extreme the peak will be on the left side of the cyclestart spike From your original calibration you can derive the wavelength associated with x1 and x2 and thereby the wavelength di erence AA M952 7 M951 across the 1024 pixels The change in wavelength per pixel will be A1024 AApixeli After you have obtained your pixel calibration data and printout for the oz lines repeat the procedure to measure the Hg and Dg lines You will need to increase the vertical sensitivity on the scope since these lines are less intense and the CCD is less sensitive to this wavelength Make sure to turn off the Averaging feature when rst locating the lines After locating the lines and parking them close to the cyclestart point increase the sweep speed and vertical sensitivity to see the lines in more detail Turn on the Averaging feature to get rid of the bounce in the display Since the lines produce a much smaller signal than the oz lines the background pixeltopixel variation of the CCD readout will visibly affect the line shapes To eliminate this background variation the REFERENCE and MATH features of the scope can be utilized After nding the lines and getting them displayed on the digital scope proceed as follows 1 Cover the slit assembly with the black paper or slide the carriage a wee bit to one side so that the lines move off screen i F Press the REF button turn Ref 1 on press the button below the Ref 1 selection on the screen and store Channel 1 to Reference 1 by pressing the SaveRecall button to the right of the knob marked EB and chosing the appropriate softkeys on the screen 9quot Remove the black paper covering the slit or move the carriage back to its original position tb Now press the MATH button and select the subtraction operation with the rst source set to Channel 1 and the second source set to Reference 1 The MATH trace red one will now display the difference between the signal and background with the pixeltopixel variation largely eliminated and the line shapes displayed much more cleanly This effect is shown in Fig 3 Make a print out of the lines for later analysis If you have time you may wish to explore whether you need to calibrate the pixel width versus wavelength for the line measurements Can you think of a reason why it might differ from that of the oz lines Copies of the scope display for both the HEXDO and HgDg lines are available in the lab so you can see what sort of data you should be able to obtain Physics 331A Experiment 2 Edit Math Definition Set ist SOUI CE t0 gggmmgg El Set 39 Dpe rato l39 to EHEHB i 1 I Z 2L1y I I Z I 39 Source to Chi 00mv f 39 460m 4 Ext39 x 204me 2 500m 500145 3300 9s g g Dual me Math Figure 3 Scope traces in the acquisition of the lines This picture shows the effect of subtracting the random pixel offsets from the signal The lowest trace near the marker Rl shows the CCD output without the lines The middle trace near the marker 1 shows the CCD output with the lines in view The topmost trace near the marker M shows the difference between the two lower traces as calculated using the Math feature On the right of the gure is the Math menu with Ch 1 subtracting Ref 1 SHUTD OWN When you are nished please move the CCD array out of the way put the viewing microscope back in place and put the covers back over the optics Shut off all lamp power supplies and electronics POST DATACOLLECTION ANALYSIS With your printouts and calibration data in hand carry out the following analyses for your report 7 Obtain estimates for the resolving power of the spectrograph by studying the widths of the hydro gendeuterium lines Estimate the width of each peak to one half of a pixel You should have four peaks to look at Multiply this number by Akpixel to obtain a minimum resolvable AA and a value for R Make sure to calculate an estimate on the random error of this value of R Calculate the theoretical value of Rgmtmg for this grating using Eq Then use this value of Rgmtmg to estimate how many pixels this resolving power would correspond to Compare this theoretical limit to the values you measure and discuss Why is this experimental value for R likely to be less than Rgmtmg What other factor do you think affects the resolving power the most Justify your choice based on your observations With this instrument how close in wavelength could two lines be and still be resolved Explain the criteria you apply to make this estimate Hint you may want to read Hecht pp 471 481 to help you think about this problem Physics 331A Experiment 2 8 Determine the mass ratio of deuterium to hydrogen from the di erence in wavelength of the oz and lines Use your printouts and calibration data to determine the di erence in wavelength AAHD E H 7 D between the Ha and DEX lines and the di erence in wavelength between the Hg and D linesi Then calculate AAHD H for each pair of lines along with an uncertainty in this ratio Use Eqs 1 2 and 3 to prove the following relationship AAHD 7 liMyMD H 7 1MHm 3 6 where M H and M D are the masses of the hydrogen and deuterium nuclei Finally use the result of Eq 6 along with your measurements to calculate the ratio of the of the masses of the hydrogen and deuterium nuclei MHMD along with its uncertaintyi You will need to use the protonelectron mass ratio mpm r3 1836151 If the nuclear mass ratio you calculate is not close to 12 try again


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