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# EXPERIMENTAL PHYSICS PHYS 2350

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This 243 page Class Notes was uploaded by Justine Nitzsche on Monday October 19, 2015. The Class Notes belongs to PHYS 2350 at Rensselaer Polytechnic Institute taught by Staff in Fall. Since its upload, it has received 7 views. For similar materials see /class/224888/phys-2350-rensselaer-polytechnic-institute in Physics 2 at Rensselaer Polytechnic Institute.

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VOLUME 18 NUMBER 8 PHYSICAL REVIEW LETTERS 20 FEBRUARY 1967 We have now seen CESR in four metals besides all the alkalis there is a good possibility for observing spin waves in at least one of them 5S Schultz and C Latham Phys Rev Letters 12 695 1965 S Schultz and M R Shanabarger Phys Rev Letters 16 187 1966 6This is a big if In our experiments leakage was typically better than 165 dB and the actual power in the main spinwave peak was N20 dB above leakage for the data presented in Fig 1 7Although we have detected the power transmitted for most purposes it is more convenient to measure the component of the transmitted magnetic field pro jected on a reference rf field and all the data shown were taken with the reference field adjusted so as to ob serve the imaginary part of the complex susceptibility ie the quantity which is called x in the usual reso nance terminology 8For our material pRTp4239 K 6000 and the ap propriate 07 W 20 Making corrections due to the finite 07 does not significantly alter any of the results pre sented T2 w 10quot6 sec 9We have used the value of 124 for mm in sodium C C Grimes and A F Kip Phys Rev 1332 1991 1963 110There are of course two roots for81 in the solu tion of the equation One of these is eliminated in that it predicts the location of the spin waves on the wrong side of the CESR 11M T Taylor Phys Rev 1231 A1145 1965 and us ing mm 124 12V F Gantmakher Zh Eksperim iTeor Fiz 42 1416 1962 translatiom Soviet PhysJETP15 982 1962H 13See Ref 1 and G D Gaspari Phys Rev 151 215 1966 14P S Peercy and W M Walsh Jr Phys Rev Let ters 1Z 741 1966 15For the range of sample thicknesses considered the diffusion time for the spin information is much less than the spin relaxation time Hence a free induction decay measures T2 The same equipment is being used for experiments at higher temperatures where the reverse relationship between the times applies MEASUREMENT OF 2612 USING THE ac JOSEPHSON EFFECT AND ITS IMPLICATIONS FOR QUANTUM ELECTRODYNAMICS W H Parker Department of Physics and Laboratory for Research on the Structure of Matter University of Pennsylvania Philadelphia Pennsylvania and B N TaylorT RCA Laboratories Princeton New Jersey and D N Langenberg Department of Physics and Laboratory for Research on the Structure of Matter University of Pennsylvania Philadelphia Pennsylvania Received 23 January 1967 Using the ac Josephson effect we have determined that Zeh 4835912 400030 MHzpV The implications of this measurement for quantum electrodynamics are discussed as well as its effect on our knowledge of the fundamental physical constants In this Letter we report a highaccuracy measurement of 2eh using the ac Josephson effect here 2 is the electron charge and h is Planck s constant When combined with the measured values of other fundamental con stants this measurement yields a new value for the fine structure constant oz which differs by 21 ppm from the presently accepted value This change in 01 removes the present discrep ancy between the theoretical and experimen tal values of the hyperfine splitting in the ground state of atomic hydrogen one of the major un solved problems of quantum electrodynamics today We also discuss the effect of this change on our present knowledge of the fundamental physical constants The phenomenon used in these experiments was first predicted by Josephson in 19621 He showed theoretically that when two weakly cou pled superconductors are maintained at a po tential difference V an ac supercurrent of fre quency V 2612 V I 1 287 VOLUME 18 NUMBER 8 PHYSICAL REVIEW LETTERS 20 FEBRUARY 1967 flows between them This equation known as the Josephson frequencyvoltage relation can be shown to follow from quite general assump tions concerning superconductivity 2 and is believed to be exact In a recent publication3 we reported an experimental test of this rela tion which verified that the frequencyvoltage ratio was equal to the then best value of Zeh to within the 60 ppm uncertainty of the mea surements and was completely independent of all of the experimental variables tested This work also demonstrated that the main experimental difficulty with using the ac Joseph son effect to make a high accuracy determi nation of Zeh was the calibration of the volt age measuring system Since then we have acquired a completely self calibrating system and have carried out a new series of measure ments with an order of magnitude improvement in accuracy In these new experiments measurements were performed on two types of junctions which show ac Josephson effect phenomena evaporated thin film tunnel junctions and point contact weak links3 The particular phenom enon used was that of microwaveinduced con stant voltage current stepsl 2 of the type first observed by Shapiro4 Such steps appear as excess dc currents in the current voltage char acteristics of Josephson junctions when they are irradiated with microwave radiation of frequency V Physically this effect arises from the presence of a microwave induced ac voltage across the junction which frequen cy modulates the ac Josephson current The constantvoltage current steps simply corre spond to zerofrequency dc sidebands The relationship between the voltages V at which these steps occur and the frequency of the ap plied radiation is 26V nhv where n is the number of the step A determination of Zeh can thus be made simply by measuring the fre quency of the applied microwave radiation and the absolute voltage at which the current steps occur No other measurements are required except those necessary in calibrating the equip ment and in this sense it is a remarkably straightforward fundamental constant experi ment The applied microwave radiation was gener ated by an X band 8 to 124 GHz oscillator with a stability of one part in 108 per hour when phase locked to a quartz crystal reference The frequency of the radiation was measured 288 to an accuracy of one part in 108 by using an electronic counter and a microwave frequen cy converter The reference time base of the counter was maintained to an accuracy of bet ter than one part in 108 by regular phase com parisons with the U S frequency standard as broadcast by radio station WWVB Fort Col lins Colorado Thus the frequency measure ment contributed negligible error 001 ppm to the measured value of 2612 The overall accuracy was limited by the voltagemeasur ing system ie potentiometer and standard reference voltage The reference voltage used was the mean voltage of a set of six standard cells calibrat ed by the U S National Bureau of Standards NBS in a constant temperature air bath This voltage was known to 1 ppm in terms of the NBS legal volt the uncertainty being an esti mate of the possible changes in the emf of the standard cells due to transporting them from NBS to our laboratory In order to obtain a value of Zeh in absolute units it is necessary to convert from the legal volt to absolute volts The present best value of this conversion fac tor is 1 NBS legal volt 1000 012 I 0000 004 absolute volts where the 4ppm uncertainty is intended to represent a 50 confidence lev e15 The potentiometer used was the Julie Research Laboratories PVP 10016 This nanovolt instru ment is selfcalibrating in that it has provisions which enable the operator to measure all fac tors which contribute to the accuracy of a volt age measurement and to make any necessary correctiOns Using techniques developed by Julie Research Laboratories7 and NBS8 the 1mV fullscale range can be calibrated with an rms uncertainty of between 3 and 4 ppm this was the range normally used since the voltages of the induced current steps rarely exceeded 1 mV The null detector used with the potentiometer consisted of a photocell am plifier and galvanometer and had a resolution of l nV In making accurate measurements of such small voltages great effort is necessary to eliminate or correct for spurious voltages in the measuring circuit In the measurements described here the effect of voltages which do not reverse when the current is reversed thermoelectric voltages for example was eliminated by measuring constant voltage cur rent steps of both polarities Voltages which VOLUME 18 NUMBER 8 reverse with current those from Ohmic sourc es for example were shown to be negligible by observing that the voltage in the measuring circuit was constant over the full range Of the zerovoltage current arising from the dc JO sephson effect1 Spurious voltages due to any rectification Of the microwaves were also shown to be negligible by observing that the measured value of Zeh was independent Of microwave power over a range Of 10 dB for a given sam ple and of 20 dB from sample to sample The results of measurments on several thin film tunnel junctions and pointcontact weak links are given in Table I The standard de viation of a set of measurements obtained dur ing one run9 on any particular junction typi cally 2 ppm is due to the lnV resolution of the null detector the stability and linearity of the potentiometer and the stability Of the thermoelectric voltages in the measuring cir cuit usually of order 100 IN Within this 2 ppm standard deviation the measured value of 2eh was found to be independent of a wide variety of experimental conditions including step number up to n 40 magnetic field from 0 to 10 G microwave frequency from 8 to 12 GHz and microwave power Allof the mea surements were carried out between 12 and 16 K For most of the current steps from which the data were obtained the voltage was con stant to within 1 nV the resolution of the null detector over the full range of the step How ever for three of the point contacts marked PHYSICAL REVIEW LETTERS 20 FEBRUARY 1967 by an asterisk in Table I the voltage was found to increase by 510 nV as the current was in creased over the range of the step It was Ob served that in higher resistance point contacts several tenths of an ohm rather than several hundredths of an ohm where the voltage vari ation was as much as 200 nV the midcurrent point of the step gave a value of 2eh equal to the average of all the data obtained on the con stantvoltage steps As the resistance was decreased the voltage variation decreased and the midpoint continued to give a value of 2eh in agreement with the constantvoltage step data Extrapolating this behavior to con tacts with 5 to 10nV variation we assume that the midcurrent point corresponds to the voltage at which the step would occur if it were constant Thus all measurements on such steps were made at this midpoint The average of all of the data in Table I weighted as the inverse square of the rms un certainties gives Zeh 2 4835912 2 00030 MHZUV or in more conventional terms he 4135 7253 0000 026x 1039quot15 J secC 1379 5293c 0000 008x 10quotquot17 erg secesu The quoted uncertainty about 6 ppm 70 con fidence level is an rms sum of all known sourc es Of error either systematic or random and includes the uncertainty in the calibration of the potentiometer the standard deviation Table 1 Summary of experimental data Junctions of the form Sn SnOX are evapo rated thin film tunnel junctions while the others are point contact weak links The table entries are in chronological order and the decreasing uncertainty in the potentiometer calibration results from improved techniques POTENTIOMETER STD CELL CORRECTED nuV UNCORRECTED nuV CORRECTION WITH TEMPERATURE WITH rms WITH STD DEV rms UNCERTAINTY CORRECTION UNCERTAINTY JUNCTION MHzuV PM PM MHzw Sn SnOSn 4838IOi OOZ 3O IO 483596i005 SnSn 4836I28iOOII 29 239 7 4835986t0032 TaTo 4836641390009 39 t 4 O5 4835973tOOZI Sn SnOSn 4836I56tooos 39 i 4 O5 4835965toozI Sn SnOSn 4836I58tOOOT 42 j 4 03 483 595511002 SnSnOPb 4838I74iOOI3 43 d 4 Io 48359621390023 NbTa 4838I95 iOOOT 42 139 4 IO 4835987iooaI ToNb33n 483 eIssiOOII 44 i 4 O5 4835975110022 ToTCI 4838I85tOOI8 46 139 4 O4 483596l ioozs SnSnOSn 483 8I94IOOOS 45 i 4 OO 483 5976 toozo WEIGHTED AVERAGE OF DATA lN TERMS OF NBS VOLT TO ABSOLUTE VOLT CONVERSION TN ABSOLUTE UNITS MHz11V FINAL VALUE FOR Zeh NBS VOLT MHZ 1VNBS 483 59710022 O OossiOOIe 48359l21390030 289 VOLUME 18 NUMBER 8 PHYSICAL REVIEW LETTERS 20 FEBRUARY 1967 of a set of measurements the uncertainty in the absolute value of the NBS legal volt and the uncertainty in transferring the NBS volt to our laboratoratory It should be noted that the standard deviation of the eight most accu rate measurements made over a period of several months is only 2 ppm an indication of the high precision of the measurements To within this 2ppm precision the measured value of Zeh is independent of the material and type of junction used A value of the finestructure constant can be derived from our value of Zeh and other directly measured quantities by use of the equa tion M 26 12 p a 0 OZ1 quotquot 41700711 110 h where c is the velocity of light R00 is the Ryd berg constant for infinite mass 39yp is the gy romagnetic ratio of the proton and upuo is the magnetic moment of the proton in units of the Bohr magneton Taking the best values for these quantities10 c 2997925gtlt108 m secquot1 1 03 ppm Reo 1097373 1 X 107 m li 01 ppm lip110 1521 032 5X 10 3i 05 ppm yp 2675 192 gtlt108 rad secquot1 T li 3 ppm and Zeh 4835 912 X 1014 Hz V li 5 ppm Eq 2 gives11 oe l 1370359100004 2 Zeh This value is 21 l 5 ppm less than the present ly accepted value derived from the finestruc ture splitting fs in deuterium as measured by Treibwasser Dayhoff and Lamb z 13 ozquot1fS 1370388 1 00006 The new value of 02 derived here entirely removes the apparent discrepancy between the theoretical and experimental values for the hyperfine splitting hfs in the ground state of atomic hydrogen This splitting vhfs has been measured to the extraordinary accuracy of 2 parts in 1011 by Crampton Kleppner and Ramsey14 The quantumelectrodynamic ex pression for the splitting which includes all theoretical effects other than the dynamic po larizability of the proton is believed to be accurate to a few ppm15 When this expression is evaluated using oz1amp5 it predicts vhjs expt vhfs theory 43 1 1 2 ppm VhfseXpt 290 The quoted errors include an uncertainty of 2 ppm in the estimate of form factors and 5 ppm in a 1fs If the theoretical expression is evaluated using a 126h it predicts V Sexpt z theory hf hf s VhfseXpt 018 ppm Although at present it is impossible to calcu late the proton polarizability exactly the best estimates indicate that it would increase Vhfsthe ory by less than 10 ppm15 Thus unless the proton polarizability is much larger than is presently believed oz 1amp8 suggests a break down of quantum electrodynamics while a 126h is consistent with both quantum elec trodynamics and a small proton polarizability The change in oz implied here is also impor tant because in the 1963 adjustment of the fun damental physical constants by Cohen and Du Mond the value ozUfs was used as an input datum Because of the pivotal role played by oz in this adjustment16 any change in oz will cause large changes in the values of the other fundamental constants In Table II we give the values of some of the more important con stants which would have resulted if a 126h had been used as an input datum in the 1963 adjustment We might also point out that with this new experimental value for 2812 a new and more reliable value for the xray wavelength conver sion factor A can be obtained Using the re cent experimental data of Spijkerman and Bearde enl7 for the voltagetowavelength conversion factor VAS we find that A 1002 067i 0000 023 Akxu based on Bearden s new definition of the x unit18 Although we believe that the results report ed here are highly reliable work is continu ing to ensure that there is no unknown system atic error in the measurements To this end experiments of the type described here are being carried out at higher frequencies 70 GHz as well as experiments involving the mea surement of the frequency of the radiation emit ted by a Josephson tunnel junction when biased to a known voltage Preliminary results from experiments of the latter type are in complete agreement with the results presented here We should like to thank Mr Loebe Julie for helpful discussions concerning the potentiom eter Mr A G McNish for arranging the cal VOLUME 18 NUMBER 8 PHYSICAL REVIEW LETTERS 20 FEBRUARY 1967 Table II Changes in some of the fundamental physical constants resulting from substituting a 128h for 011 f8 in the input data of the 1963 adjustment N is Avogadro s number and m is the rest mass of the electron The numbers in parentheses are the one standard deviation errors in ppm Value given Value implied by 1963 by this Change Quantity Units adjustment measurement ppm a1 13703884 13703593 21 e 10 19 c 1602 1013 1602 2013 63 10 10 esu 4802 9813 4803 2813 h 10quot34 J sec 6625 5924 6626 2824 105 me 1031 kg 9109 0814 9109 6514 63 N 1026 kmole 1 6022 5215 6022 1415 63 ibration of the standard cells Dr E Richard Cohen and Dr Jesse W M DuMond for their interest and encouragement Professor S A Bludman and Professor D J Scalapino for helpful discussions and Mr A Denestein for his excellent technical assistance A contribution from the Laboratory for Research on the Structure of Matter University of Pennsylvania covering research sponsored by the National Science Foundation and Advanced Research Projects Agency TPart of this work was performed While the author was at the University of Pennsylvania and part While at RCA Laboratories 1B D Josephson Phys Letters 1 251 1962 Ad van Phys 14 419 1965 2P w Anderson Rev Mod Phys as 298 1966 Progress in Low Temperature Physics edited by C J Gorter North Holland Publishing Company Amster dam 1964 Vol IV 3D N Langenberg W H Parker and B N Taylor Phys Rev 136 186 1966 4S Shapiro Phys Rev Letters 11 80 1963 S Sha piro A R Janus and S Holly Rev Mod Phys 36 223 1964 5F K Harris Electric Units to be published and private communication 6L Julie An Unusually Accurate Universal Poten tiometer for the Range from 1 Nanovolt to 10 Volts unpublished 7L Julie A Ratiometric Method for Precise Cali bration of Volt Boxes to be published 8R F Dziuba and T M Souders A Method for Cali brating Volt Boxes with Analysis of Volt Box SelfHeat ing Characteristics IEEE International Convention Record March 1966 Pt 10 9A typical run might consist of 40 separate measure ments on about 8 different steps 101 R Cohen and J w M DuMond Rev Mod Phys 91 537 1965 Throughout this Letter all uncertainties correspond to one standard deviation unless otherwise stated The uncertainties for VP and 28 h do not include the uncer tainty in the conversion factor from NBS as main tained electrical units to absolute units because the conversion factor for each of these quantities is essen tially the same and enters Eq 2 in such a way as to cancel out Thus we are able to completely bypass the problem of the relationship between NBS legal units and absolute units 12S Triebwasser E S Dayhoff and W E Lamb Jr Phys Rev 9 98 1953 13The recent measurements of the hyperfine splitting in muonium by W E Cleland gt al Phys Rev Let ters E 202 1964 after correction by M A Ruder man Phys Rev Letters 1 7 794 1966 as well as the measurements of R T Robiscoe and B L Cosens of the Lamb shift in H Phys Rev Letters 11 69 1966 indicate that oz 1HS may be in error Also see Ref 16 1quot S B Crampton D Kleppner and N Ramsey Phys Rev Letters 11 338 1963 15For an up to date review of the theoretical situation and the original references see S D Drell and J D Sullivan Phys Rev to be published 16J W M DuMond Z Naturforsch 21a 70 1966 17J J Spijkerman and J A Bearden Phys Rev 4 A871 1964 18J A Bearden gt a X RaXWavelengths Report No NYO 10586 Clearing House for Federal SCientific and Technical Information Springfield Virginia 1964 291 Experimen ral Physics PHYS 2350 Prof Gary Adams SCIW15 adam59r39piedu Office hours ExPhys hr39s or39 10am Wed and Friday Ex Phys 2005 1 Class Information Class hours and loca rions REGULAR LECTURE TWF 121250PN LAB M M 8001050AM and 200250PM LAB R R 8001050AN and 200250PM Labs will be held in 563614 Web sil39e wwwrpiedudep rphysCour39sesPHYSZ350 Ex Phys 2005 2 AlThough The sTaTed lab hours ToTal 4 hours per39 week you should anTicipaTe ThaT you will need 6 hours per39 weeks To do The experimenTs well You should arrange wiTh your39 lab par39Tner39s To meeT for39 The r39equisiTe hour39s Required Tebeook ExperMenfa Physcs noTes by James NapoliTano IT can be downloaded from The web siTe Grading mosle wr39iTe ups see syllabus Ex Phys 2005 3 La re assignmen r policy always good To finish Academic hones ry You are expec red To meaningfully par ricipa re in all labor39a ror39y experimen rs If we de rer39mine Tha r a s ruden r is simply using da ra from ano rher39 s ruden rs we will give you a zero for Tha r lab You are encouraged To work wi rh your39 lab par39Tner39 on analysis bu r The ac rual wr39i re up mus r be your39 own Experimen rs for Spring 2001 A Vol rage dividerfiller E shor39T Ex Phys 2005 4 B Coun ring s ra ris rics C shor39T C Dielec rr39ic func rion E D Resis rivi ry by induc rive decay E E Tempera rur39e coefficien r of r39esis rivi ry E F A romic spec rr39oscopy G G Johnson noise E G Faraday effec r G H Radioac rivi ry C J Posi rr39on annihila rionC K Comp ron effec r C L ULI mechanics spring Ex Phys 2005 5 M ULI mechanics pendulum N ULI r39o ra rional iner ria Ex Phys 2005 6 Wr39i re ups The quotminilabquot wr39i re up M The real Time logbook S ra remen r of goals S ra remen r of planned approach Descrip rion of equipmen r model and serial if possible ske rch layou r Descrip rion of logic wha r was measured Raw da ra Simple plo rs Obser39va rions or39 ideas Ex Phys 2005 7 Par r 2 a separa re wri reup brief descrip rion of experimen ral procedure analysis and discussion final plo rs and graphics error analysis conclusions The quots randardquot wri reup Add Ti rle and Abs rrac r o Rewri re Par r 1 above neale in clear English with complel39e senl39ences Ex Phys 2005 8 ParT 2 same as for quotminilabquot wriTe up Final formal reporT Add a secTion wiTh inTroducTory ideas and basic Theory LaboraTory Sa eTy 0 Review safeTy issues for each experimenT wiTh your parTners and The TA or me before you sTarT Do noT work alone in The laboraTory 0 Think before you do Ex Phys 2005 9 c When in doub r ge r ou r And war39n o rher39s of risk Specific issues for 3614 lab Pease use ve 39sf beaw as a safe fy checkIs for each experimenf as you sfaquotf if Elec rr39ical shock vol rage gt20V can br39idge skin r39esis rance or39 air39 gaps cur39r39en r kills Wa rch for charge on capaci ror39s cur39r39en r in induc ror39s magne rs Ex Phys 2005 10 Keep one hand in your pocke r TEST VOLTAGES AND THINK Explosionimplosion Gas con rainers cryogens CRT39s ligh r bulbs vacuum Tubes Asphyxia rion from rapid boiling of cryogenic gases THINK Poisoning by inges rion or Through skin lead bricks Hg swi rches Be windows dyes and organics radioac rive sources Ex Phys 2005 11 NO FOOD IN THE LAB WASH YOUR HANDS WHEN YOU LEAVE Op rical damage laser beams Hg discharge lamps AVOID LOOKING DOWN ANY BEAM DON39T EXPOSE OTHERS HOTcold THINK Radioac rivi ry REAL TIME MONITOR GEIGER WEAR EXPOSURE BADGE WASH YOUR HANDS WHEN YOU LEAVE THE LAB BE FAMILIAR WITH LIMITS OF EXPOSURE Ex Phys 2005 12 RadiaTion saieT Exposure This is The ionizaTion produced per uniT of maTTer The sTandard uniT is The RoenTgen R which is The quanTiTy of X rays ThaT make one elecTrosTaTic uniT of ionized charge in 1 cm3 of air Absorbed Dose This is The energy deposiTed per uniT mass The old uniT is The rad defined To be 100 erggram The SI uniT is The Gray Gy or 1 Jkg100 rad Ex Phys 2005 13 To conver r be rween dose and exposure you need To know how much energy in a par ricular ma rerial produces so much ioniza rion For air if Takes on The average for elec rrons 337 eV of deposi red energy To crea re an ion pair The conversion fac ror be rween exposure Roen rgen and dose rad is abou r 1 The rem is a uni r of damaging dose To humans and includes a qualify fac rorquot which is 1 for gamma and x rays and 10 for alpha par ricles see no res handou r for more informa rion Ex Phys 2005 14 A useful conversion for gamma rays 1 Curie of gamma source produces 13 rem per hour a r a dis rance of 1 me rer So Doserem13ACithre d r070 AacTiviTy in Ci r disTance in me rers 139139ime in hours aaTTenuaTion coefficien r for shielding dz rhickness of shielding Ex Phys 2005 15 Good exger39imenTal Technigues Use fhe sf beow as a checkIs as you proceed fhrouqh each experiment Read The book on ThaT experimenT before you come To The fir39ST lab class Wr39iTe a shor39T lisT of goals inTo your39 noTebook Think abouT The logical experimenTal approach To achieving The goals Think abouT and make a lisT of safeTy issues for your39 experimenT EsTimaTe r39adiaTion dose Ex Phys 2005 16 Familiarize yourself wiTh The equipmenT PracTice Taking daTa Develop a plan Take daTa while TesTing To see how robusT or reproducible The sysTem is Assess uncerTainTy in each parameTer and how iT affecTs your conclusions Record daTa in your noTebook in The raw form eg On an oscilloscope record The scale seTTings and Then The number of minor divisions You may also record calculaTed Ex Phys 2005 17 parameTers aT The same Time Assess your reading error while you are making readings Think abouT how you could change The measuremenT approach or condiTions and how you expecT ThaT change To influence The measuremenT or resulT 39TesTyourideas Ex Phys 2005 18 Uncer rain ry Sys rema ric and Random gt Random uncer rain ry arises from The fundamen rals of a measur39emen r or39 ins rr39umen r The measur39emen r Typically sca r rer39s abou r an average value gt Sys rema ric uncer39Tain ry is common To a se r of da ra and depends on exper39imen ral me rhod Ex Phys 2005 19 EsTimaTion of uncerTainTy a A sensible fracTion of The leasT significanT digiT on your measuring device b Analysis of The sTaTisTics from a seT of measuremenTs Common Techniques To reduce sysTemaTic uncerTainTx Reverse or swiTch leads SwiTch measuremenT devices CalibraTe againsT a known sTandard eg 60 Hz line volTage Ex Phys 2005 20 RepeaT measuremenTs Have parTner repeaT measuremenTs Avoid Taking daTa in monoTonic incremenTs eg Always increasing or always decreasing volTage Measure raTios or differences SeT up The experimenT so iT only gives you differences PloT daTa in your noTebook or on your lapTop compuTer as you go To look for anomalies Scale The experimenT ie double lengTh of wire change area of wire Ex Phys 2005 21 Probabili ry for random errors Gaussian or Normal dis rribu rion ParameTers are mean vcdue widTh and To rcd probabiih y WidTh usuaHy given as The sTandard deviaTion 5 Ex Phys 2005 22 Normal Gaussian function sigma1 area1 xlsigma Ex Phys 2005 23 ProgagaTion o UncerTainTies How does The uncerTainTy in a parTicular measuremenT affecT a calculaTed resulT 1 BruTe force meThod Plug your uncerTainTy esTimaTe direchy inTo The TheoreTical predicTion and calculaTe The difference 2 Expansion approximaTion Taylor expand your funcTion abouT The measured poinT Carefully add uncerTainTies TogeTher Ex Phys 2005 24 Known func rion of a single variable Taylor expansion5q 5x for rela rively small uncer rain ries Example GAV3 where ATheoreTical cons ran r and Vmeasured quan ri ry 3 50 3 0 6V3AV026V 3A V0 5V3GO 3V H0 V0 V0 zg Go V0 Known func rion of several variables dq dq 5 x x05x 5qy d y 5y Correla red uncer rain ry x and y are rela red Ex Phys 2005 25 carefully add uncer rain ries Uncorrela red uncer rain ries add in quadra rur39e5q2 561 M Example Wha r if A in The problem above were also an independenle measured quan ri ry 8 6 we we 3V VV0 3A AA0 V0 A0 502 2 35V 2 g 2 03 V0 A0 5 is called The frac rional uncer rain ry 2 502 Ex Phys 2005 26 STaTisTical Analysis CompuTaTion of uncer39TainTy from several measuremenTs of The same quanTiTy Suppose you Take n measuremenTs of a quanTiTy x In previous classes you used The following definiTions The m value of x as fltxgtlixi 7 i1 The sTandard deviaTion of x as ax m2 1ltx2gt ltxgt2 Why did you do This oTher39 Than ThaT iT was a def i ni Ti on Ex Phys 2005 27 The loqic behind sTaTisTical analysis 1 To find The besT esTimaTe for a par39ameTer39 in a fiT To a seT of measuremenTs 2 To TesT wheTher39 The funcTion you chose is appropriaTe hypoThesis TesTing NoTe each daTa poinT in a seT of measuremenTs is usually assumed To be Gaussian disTribuTed In ThaT case hypoThesis TesTing can be based on The magniTude of Ex Phys 2005 28 The Tool 7 chi squared We define 2 yi fx2 Z 52 5132 where X and y are measured 6y is The uncer rain ry in a measuremen r of y and 67 is The uncer rain ry in fcompu red from The uncer rain ry in X A reasonable approach To finding bes r fi r parame rers is To minimize f Ex Phys 2005 29 Example Bes r fi r To a cons ran r The average The func rion for a fi r fxA 2 quot yi AY Z of 0 2 1 2xi A 2x1 2A M2 02 ZZ O solveforA A Eweof 2003 Ex Phys 2005 30 The quanTiTy wi is called The weighT of a measuremenT A Then is jusT The weighTed average The uncer39TainTy in The weighTed average is given by 0A For39 a Gaussian disTr39ibuTion of measuremenTs 0A is known as The sTandar39d deviaTion 1 n For39 equal uncer39TainTies A ZZX O39A O39iVn1 i1 Ex Phys 2005 31 Some useful MATLAB functions sumx meanx s rdx where x is an array of The measured values Ex Phys 2005 32 We go Through a similar minimizaTion process solving Two simulToneous equaTions for A and B To find The slope and inTercepT of a sTraighT line fiT yzABX To The daTa Ex Phys 2005 33 Assuming 03 is independen r of fund Tha r 0X is negligible A Z XZyJZXZXZXZyi nZ in2 B quotExtyoZXZXzyi quotEXQZXZY 02 032x A nfogt in2 02 2quot y 03 nfo in2 Ex Phys 2005 34 92 9002 39s qd 39xa 111 p2114612MA440nb2 40 Ip2114E51211unquot 141 s 5114L 21oN 39DDp 2141 01 111 01 114011 noA IngwouAlod 2141 Lo 42pJo 2141 s u 24214M u A x 11140dzd MOHDUHJ 8V1va 14950 39qnl 15414 JnoA 11 42dDd uo uy u4n4 3917 U424qo4d 6 39149 1 pun I su424qo4d 9 39149 cumuloan U2UJU5SSV see hw192m in Ma rlabexamples Ex Phys 2005 36 quot6 This is a program To demons rr39a re polynomial quot6 regression and chiquot2 ideas clear39 x78 8O 87 84 8180 82 8O 87 9185 8185 74 86 77 89 87 91 90 y65 56 7173 72 6o 67 66 63 74 64 66 66 58 66 59 78 76 78 68 hold on ppolyfi rxy1 xf70195 yfiTppolyvalpxf Ex Phys 2005 37 yfiTpolyvalpx plo rxy39 39 plo rxfyfi rp xbarmeanx ybarmecmy covarizcovxy cor39r39cocor39r39coefxy Chi5qsumyyfif 24 2 rm A u gag cEJJia g g39gc s 2 2quot purl at 3 b u 01 R33 prin r deps covarexeps Ex Phys 2005 38 p 09680 137726 xbar39 837500 ybar39 673000 covariz 234605 227105 227105 419053 corr39co 10000 07243 07243 10000 chsq 236559 Ex Phys 2005 39 No re To Tes r our39 hypo rhesis de rer39mine goodness of HT we can use fdegr39eesoffr39eedom fpoin rs parame rer39s Ex Phys 2005 40 80 50 70 75 80 85 90 95 Fits 1390 nonlinear functions of The free parameters Fir39s r don39T Try To linearize ins read Search for bes r fi r parame rer39s MATLAB fns fmins do numerical minimiza rion See 39generalji r rernm39 for example Ex Phys 2005 42 Wha r does x2 mean Using x2 as a measure of goodness of fitquot The area under a x2 curve in regra red from x2 0 gives The probabili ry Tha r repea ring a measuremen r will produce a smaller value x2 has an expec red average value of v DOF for large number of poin rs Any rhing much smaller or larger is improbable Large x2 is an indica rion Tha r you have chosen The wrong func rion for your fi r Small X2 indica res Tha r you39ve es rima red your error incorrec rly Ex Phys 2005 43 The x2 dis rribu rion 1 Po xgtrszzltugtmz Isuquote7du 2 2 PM 2 x v5 Probability Density Function mm P a u Ex Phys 2005 44 For large vgt30 JEW m is Gaussian dis rr39ibu red wi rh a mean of zero and variance 62 equal To one No re and 6 increase wi rh v Ex Phys 2005 45 PracTical x2 analysis 1 Make esTimaTes or39 calculaTe The sTandar39d deviaTion of your measured quanTiTies a in counTing experimenTs The uncer39TainTy is The square r39ooT of The counTs J si b in measuremenT experimenTs eg voleeTer39 i r39epeaT measuremenTs and compuTe sTandar39d deviaTion ii esTimaTe uncer39TainTy from leasT significanT digiT Ex Phys 2005 46 iii observe var39iaTion or39 dr39ifT similar To r39epeaTed measuremenTs iv keep in mind ThaT The meThod of sampling can skew The daTa v consTr39ucT a daTa Table x dx y dy 2 dz 1 001 2 0015 201 14 2 001 3 0017 250 16 Ex Phys 2005 47 2 Decide on a fiT funcTion To TesT a NoTe The number of parameTers you use in your fiT funcTion eg z fxy abxc dy has 4 parameTers and Two variables b ConsTrucT chi squared 12 2 Zi fXy2 H 5212 5x12 5y y 8x 3 Compare chi squared To The disTribuTion funcTion quotchisquaredquot for The given number of daTa Ex Phys 2005 48 poinTs minus The number of parameTers For larger poinT numbers you can compare To a gaussian funcTional form You can Then describe The probabiliTy ThaT a repeTiTion of The experimenT would give you a smaller or larger chi squared value If p is ouTside of 010 O9 Then iT is likely ThaT someThing is noT righT abouT your daTa or analysis Ex Phys 2005 49 Impor ran r Probabili ry Dis rr39ibu rions These are very impor ran r for coun ring s ra ris rics ie pho ron coun ring or39 radioac rivi ry Ex Phys 2005 50 Binomial disTribuTion If an evenT has The probabiliTy of occurrence 0 for a single measuremenT Then The probabiliTy ThaT iT will occur v Times in n Trials is b V Z V 1 Pl meanunp variance 02npJp NoTe ThaT This is also The expansion for xy Ex Phys 2005 51 Example Consider a normal six sided die Toss iT 20 Times WhaT is The probabiliTy ThaT The number 5 will Turn up 9 Times The probabiliTy of a 5 on any single Thr39ow is p16 20 1 9 5 b203169 911gj g 00048 Ex Phys 2005 52 Poisson dis rribu rion Limi r of binomial when 0 becomes small and n becomes large such Tha r Hp cons ran rw u Pll V V e Variance 02w All random arrival coun ring experimen rs can be recas r as Poisson problems eg Wha r is The probabili ry Tha r a coun r occurs in 103910 sec Ex Phys 2005 53 The Poisson distribution is described by one Examples of Poisson distribution 02 018 016 014 012 probability 0 008 006 004 002 0 0 10 20 30 40 50 60 parameter the average value Ex Phys 2005 54 Gaussian dis rribu rion gt Approxima rion To binomial for large n gt and any 0 gt Good approxima rion To Poisson for large ugt10 average values Take yaZ gt Useful for39 approxima ring con rinuous func rions gt S randar39d Normalquot Gaussian func rion has 1 X v2 ar39ea 1G 0V 0 e p 202 wher39e meanw var39iancezoz Ex Phys 2005 55 Normal Gaussian function sigma1 area1 045 xlsigma Some in regr39als ITIG PgtltltT O O 5 38 1 68 Ex Phys 2005 56 2 954 3 997 References 1 Schaum39s Ou rlines Ma rhema rical Handbook 2 Lyons S ra ris rics for nuclear and par ricle physicists 3 Beving ron Da ra reduc rion and error analysis for The physical sciences Ex Phys 2005 57 Covariance and Correla rion Le r39s revisi r error propaga rion Earlier we said 302 302 5A2 M GA A0 5V2 Le 602 Bu r This was only for uncorrela red errors In general ExPhys2005 58 1 n 2 0 7Zqq n 1 Z 1 1 1 n 6 11 1121 2 2 0223 61 02 a q 022 039 q 3x x 3y y 3x 3y xy where BC 2 ax x1 xgtyl w a 2 xy n li 1xifyii For uncorrela red errors ny averages To zero Ex Phys 2005 59 Ar39e Two measuremen rs cor39r39ela red wi rh one ano rher39 Should I add uncer39Tain ries individually or39 in quadr39a rur39e Tes r for linear cor39r39ela rion r 2061 7631 37 axy lzm ozzm ygt2 my r is a number bel39ween 1 and 1 r39 oxy 0 if x and y are uncor39r39ela red oxy is called The covariance of x and y Ex Phys 2005 60 No re ltst is easy To measure bu r hard To es rima re and hard To in rer39pr39e r Ex Phys 2005 61 Probabili ry dis rr39ibu rion for quot39 The probabili ry PIr39gtr39o Tha r n measuremen rs of Two uncorrela red variables xy would produce a cor39r39ela rion coefficien r in percen r wi rh M gt r39o is 0 1234567891 n 3 100 94 87 81 74 67 59 51 41 29 6 10085705643 312112 61 10 100 78 58 40 25 14 7 2 05 01 OOOO 201006740208 2 05010 0 Ex Phys 2005 62 50 10049163 04010 0 0 0 ol Adding uncer rain ries for correla red measuremen rs 1 0xy NZW 7001 y 2 2 2 a6 2 a6 2 a6 861 x y 2axayjaxy Ex Phys 2005 63 Spreadsheef example s ruden rgl 92 d9191 d9292 dgld92dgldgld92d92 glbar gaar 1 90 90 326 129 4205 1063 1664 2 60 71 26 61 159 676 3721 3 45 65 124 121 150 1538 1464 4 100 100 426 229 9755 1815 5244 5 15 45 424 321 1361 1798 1030 6 23 60 344 171 5882 1183 2924 7 52 75 54 21 1134 2916 441 8 30 85 274 79 216 7508 6241 9 71 100 136 229 3114 185 5244 10 88 80 306 29 8874 9364 841 sums 574 771 3675 7920 2797 avgs 57 77 4 1 Ex Phys 2005 64 Ex Phys 2005 0781 65 Test for grade correlation 120 100 o o o 80 o N o w o E 60 o 0 4o 7 2O 0 I I I 0 2O 4O 60 80 100 120 Grade 1 Ex Phys 2005 66 Nonlinear curve iTTing examgle I have posTed an example of a MATLAB program ThaT uses The funcTion 39fmins39 To fiT a Gaussian plus background curve To noisy daTa The main program fiTTingrouTinem is used To inpuT The daTa inpuT The parameTer guesses call fmins and ploT The resulT fmins is a funcTional minimizaTion subrouTine ThaT minimizes The named funcTion in our case Ex Phys 2005 67 chi 2exmquot using The parameTers ThaT you pass To il39 We are minimizing chi squared which is described in The program chi2exmquot You can specify any funcTion you please chi2exmquot in Turn calls The funcTion ThaT we are using for The fiT In our case This is named fnfiTmquot chi 2exmquot passes The fiT parameTers back To The main program Through fmins The parameTers are conTained in The array pars Ex Phys 2005 68 Ex Phys 2005 69 Eundamen ral Sources of Noise Brownian mo rion KEkT2 for each degree of freedom Transla rion in gases and liquids vibra rions in solids 1 Sho r noise due To fluc rua rions in curren r due To s ra ris rical fluc rua rions in elec rron densi ry 2eAv112 V i 2 Johnson noise due To Thermal fluc rua rions in average elec rron veloci ry resul ring in ne r vol rage Ex Phys 2005 70 across any resis ror 6V 2 4kTRA V 2 3 1f noise There is no known single fundamen ral source bu r noise wi rh 1f spec rrum is observed in many sys rems from Transistors To Thin me ral films Ex Phys 2005 71 Other Sources of Noise 60 Hz pickup harmonics Radio and Television signals Acous ric pickup Coun ring s ra ris rics DC pickup eg capaci rive coupling To your39 body 9165 er Ex Phys 2005 72 Noise Reduction Techniques I THINK about where the noise might come from in your experiment before acting a checklist 1 Twist and shield wires 2 Ground to a single point to avoid current loops 3 EM Shielding Faraday cage 4 Vibration isolation 5 Use RLC or op amp circuits to select desired frequency range suppress undesirable signals Lock in amplifier Ex Phys 2005 73 6 Average 7 Op rimize experimen ral me rhod 8 Design your sys rem To measure differences or ra rios direc rly 9 GaTed deTecTion 10 Smoo rh your da ra wi rh a s ra ris rical func rion Be careful Ex Phys 2005 74 Some recommended sources Mi Meiissinos Experimem ai Physics HorowiTz and Hi The ArT of EiecTronics Tayior An InTroducTion To Error Analysis The STudenT Guide To MATLAB Squires PracTicai Physics Unusuaireferences Applica rions noTes or TuToriais in manufacTurer39s caTaiogs STanford Research SysTems signal coiiecTion lock ins gaTed counTers boxcarinTegraTor PrinceTon Applied Research lock in boxcar Oriei opTics NewporT op rics mechanical componem s Meiies GrioT opTics Ex Phys 2005 75 Canberra counTers Timers ionizing radia rion deTecTor39s KeiThiey voiTmeTer39s ammeTers eiecTromeTers Buyer39s Guides Physics Today Laser Focus Ex Phys 2005 76 fi rTing r39ouTinem hold off clear ch reseT load data load exlTxT o a load data into 1D arrays channelex11 inTensiTyex12 The fitting function is stored in fnfitm parslbackground pars2amplitude of Gaussian pars3center of Gaussian pars4width of Gaussian Start least squares fitting procedure by guessing starting points for the parameters pars10 100 80 10 guess fifparfmins39chi22x39par39s0channelin rensiTy minimize the output of the program chi2ex using the program fnfit for the function fnfitm is called from chi2exm fitpar will print on the screen as an array compute the fitted function fT1fnfiTchcmneifiTpar39 o a report chi squared Ex Phys 2005 77 chi2chi22xfifpar channei inTensiTy Ex Phys 2005 78 CHIZEXm func on chisqrchiZexpar39schanneiinTensiTy chisqrsuminTensiTy fnfiTchanneipar39squot2 FNFITm Gaussian plus polynomial background function func on ygfiTchannelpar39s ypar3951par39s2exp channe par39s32par39s4quot2 Ex Phys 2005 79 2000 1500 1000 inlensity 500 0 fifpar 5197 9909 1018 199 Ex Phys 2005 80 200 9 mass on spring In Fig x we show The log of The ampiiTude exTr39emes of The moTion pioTTed as a funcTion of Time A besT fiT sTr39aighT iine is also pioTTed DeviaTions from exponenTiai decay ar39e ciear iy visibie One way To quanTify The deviaTion is To compuTe The r39ooT 0 50 100 wigs 200 250 300 mean square deviaTion log10 position m 1 2 from The fiT a n 2 2xi xfit The n Z is because we have used Two parameTer39s To fiT The daTa For The exponenTicd fiT The sTandard deviaTion is 25 mm Ex Phys 2005 81 1amplitude m391 200 9 mass on spring Ex Phys 2005 I I 100 150 time s 82 Inverse ampliTude from figure 3 pioTTed againsT Time for comparison wiTh equaTion 19 The besT fiT slope is 035 539 1ml The sTandar39d deviaTion is 14 mm almosT a facTor39 of Two smaHer39 Than The exponenTial fiT EXPERIMENTAL PHYSICS Notes for Course PHYS235O Jim Napolitano Department of Physics Rensselaer Polytechnic Institute Spring 1999 Preface These notes are meant to accompany course PHY82350 Experimental Physics7 for the Spring 1999 semester They should make it much easier for you to fol low the material and to be better prepared for the experiments The course will riot cover everything in these notes7 but with some luck the notes will continue to be a useful reference for you The text is organized into two types of major sections7 namely Chapters and Eaiperimerits7 so that they follow in a more or less logical order As much as possible7 the Experiments only rely on material in preceding chapters There is no index7 but hopefully the table of contents will be good enough for the time being Thanks to helpful comments from many students and faculty7 this has all gone through a number of revisions which I hope have made the material more useful and more clearly presented In the latest version7 l7ve reformatted everything into ETEXQe7 the new MEX standard For the time being7 l7ve removed the explicit distinction between Experiments77 and Chapters but the references should still be clear My apologies for any mistakes l7ve made which I didn7t nd in time This change allows me to use what I think are more a more clear postscript font Special thanks to Prof Peter Persans for his comments7 and for adding the JarrelleAsh spectrometer to the laboratory for the Atomic Spectroscopy measurement l7ve updated the Procedure77 section of that experiment to include a description of this instrument Credit also goes to Peter for the expanded appendix giving a quick review of MATLAB commands Please give me any comments you might have on these notes7 particularly if you see ways in which they may be improved Thanks for your help Jim Napolitario7 January 37 1999 Values of Physical Constants The following table of fundamental constants is taken from the Review of Particle Properties 7 published in Physical Review D 17 v50 1994 The uncertainties in the values are very small and can be neglected for the exper iments in this book Quanitity Symbol Value Speed of light in vacuum 0 299792458 msec Planck7s constant h 66260755gtlt10 34 J sec h27T 65821220gtlt10 22 MeV sec Electron charge 6 160217733gtlt10 19 Coul he 197327053X10 13 MeV H1 Vacuum permittivity 60 8854187817 gtlt 10 12 Fm Vacuum permeability MO 47139 gtlt 10 7 NA2 Electron mass m5 051099906 MeVCZ Proton mass mp 93827231 MeVCZ Deuteron mass md 187561339 MeVCZ Atomic mass unit u 93149432 MeVCZ Rydberg energy thoo 136056981 eV Bohr magneton MB 578838263gtlt10 M MeVT ell27715 Nuclear magneton MN 315245166gtlt10 M MeVT 6712771p Avogardro constant NA 60221367gtlt1023 atomsmole Boltzmann constant k 1380658gtlt10 23 JK Contents 1 Data Taking and Presentation 1 11 Your Log Book 2 12 Common Sense 3 121 Use Redundancy 3 122 Be Precise7 But Dont Go Overboard 4 123 Measure Ratios 5 124 Avoid Personal Bias 6 13 Tables and Plots 7 131 Tables of Data and Results 7 132 Making Plots 9 14 Using Computers 11 141 Programs for the PC 12 142 Programs on RCS 13 143 MATLAB 14 iii CONTENTS 15 Formal Lab Reports 18 16 Exercises 19 Basic Electronic Circuits 21 21 Voltage7 Resistance7 and Current 22 211 Loop and Junction Rules 23 212 The Voltage Divider 25 22 Capacitors and AC Circuits 25 221 DC and AC circuits 27 222 lmpedance 30 223 The Generalized Voltage Divider 31 23 lnductors 33 24 Diodes and Transistors 34 241 Diodes 35 242 Transistors 37 25 Exercises 38 Common Laboratory Equipment 43 31 Wire and Cable 43 311 Basic Considerations 44 312 Coaxial Cable 45 313 Connections 47 CONTENTS V 32 DC Power Supplies 48 33 Waveform Generators 49 34 Meters 50 35 Oscilloscopes 51 351 Sweep and Trigger 52 352 Input Voltage Control 53 353 Dual Trace Operation 54 354 Bandwidth 54 355 XY Operation 55 36 Digitizers 55 3 61 ADO7s 55 362 Other Digital DeVices 57 363 Dead Time 57 37 Digital Oscilloscopes 58 371 The LeCroy 9310 Digital Oscilloscope 59 38 Computer lnterfaces 61 39 Exercises 64 4 Experiment 1 The Voltage Divider 67 41 The Resistor String 67 42 AddingaCapacitor 69 Vi CONTENTS 43 Response to a Pulse 72 Experiment 2 The Ramsauer Effect 73 51 Scattering from a Potential Well 74 511 Transmission past a One Dimensional Well 74 512 Three Dimensional Scattering 78 52 Measurements 80 521 Procedure 82 522 Analysis 84 53 Advanced Topics 85 Experimental Uncertainties 89 61 Systematic and Random Uncertainties 90 62 Determining the Uncertainty 92 621 Systematic Uncertainty 92 622 Random Uncertainty 93 623 Using MATLAB 94 63 Propagation of Errors 95 631 Examples Fractional Uncertainty 98 632 Dominant Uncertainty 100 64 Exercises 101 CONTENTS vii 7 Experiment 3 Gravitational Acceleration 105 71 Gravity and the Pendulum 105 711 Principle of Equivalence 109 72 Measurements and Analysis 110 8 Experiment 4 Dielectric Constants of Gases 115 81 Electrostatics of Gases 116 82 Measurements 121 821 Procedure 123 822 Analysis 124 83 Advanced Topics 126 9 Statistical Analysis 129 91 The Mean as the Best Value 130 92 Curve Fitting 132 921 Straight Line Fitting 132 922 Fitting to Linear Functions 135 923 Nonlinear Fitting 138 924 X2 as the Goodness of Fit 139 93 Covariance and Correlations 140 94 Distributions 144 941 The Binomial Distribution 145 viii CONTENTS 942 The Poisson Distribution 149 943 The Gaussian Distribution 151 95 Data Analysis With MATLAB 153 96 Exercises 156 10 Experiment 5 Resistivity of Metals 161 101 Resistance and Faraday7s Law 162 1011 Resistance and Resistivity 162 1012 The Eddy Current Technique 166 102 Measurernents 169 1021 Procedure 171 1022 Analysis 173 103 Advanced Topics 175 11 Light Production and Detection 177 111 Sources of Light 179 1111 Therrnal Radiation 179 1112 Discrete Line Sources 181 1113 Lasers 182 112 Measuring Light lntensity 184 1121 Photographic Filrn 185 1122 Photornultiplier Tubes 186 CONTENTS ix 1123 Photodiodes 192 113 Exercises 194 12 Experiment 6 Atomic Spectroscopy 197 121 Energy Levels of the Hydrogen Atom 199 1211 Corrections 203 122 Measurements 206 1221 Procedure Baird Spectrograph 207 1222 Procedure JarrelliAsh Spectrometer 212 1223 Analysis 216 123 Advanced Topics 222 13 Noise and Noise Reduction 227 131 Signal and Noise 228 1311 Example Background Subtraction 229 132 Kinds of Noise 232 1321 Shot Noise 233 1322 Johnson Noise 235 1323 1f Noise 236 133 Noise Reduction Techniques 237 1331 Frequency lters 237 1332 Negative Feedback and Operational Ampli ers 239 X CONTENTS 1333 The Lock In Ampli er 244 134 Exercises 247 14 Experiment 7 Johnson Noise 251 141 Thermal Motion of Electrons 252 142 Measurements 255 1421 Procedure 258 1422 Analysis 262 143 Advanced Topics 263 1431 Analysis of Traces 264 1432 Frequency Spectrum 264 1433 Circuit Modi cations 267 15 Experiment 8 The Faraday Effect 269 151 Magnetically lnduced Optical Rotation 270 1511 Electromagnetic Waves and Polarization 270 1512 Light Propagation in a Medium 275 1513 The Faraday Effect 276 152 Procedure and Analysis 279 1521 Polarization Calibration 280 1522 Applying the Magnetic Field 281 1523 Using the Lock In 284 CONTENTS 153 Advanced Topics 16 Experiment 9 Nuclear Magnetic Resonance 161 Nuclear Magnetism and Precession 162 163 Advanced Topics Measurements 1621 Equipment Settings and Parameters 1622 Procedure and Analysis 1631 Spin Relaxation Times 1632 Magnetic Moments of Nuclei 17 Elementary Particle Detection 171 173 Ionizing Radiation 1711 Charged Particles 1712 Photons and Electrons 1713 Neutrons 1714 Radiation Safety Kinds of Particle Detectors 1721 Solid Angle 1722 Gaseous Ionization Detectors 1723 Scintillation Detectors Pulse Processing Electronics Xi 285 288 292 295 300 303 303 303 305 306 308 311 315 316 319 319 321 324 331 xii CONTENTS 1731 Ampli ers 331 1732 Discriminators and Single Channel Analyzers 332 1733 Processing Logic Signals 333 174 Exercises 334 18 Experiment 10 Radioactivity 337 181 Nuclear Decay 338 182 Measurements 343 1821 Particle Counting Statistics 344 1822 Detecting Radiation 346 1823 Half Life Measurements 348 19 Experiment 11 Positron Annihilation 359 191 Correlated Pairs of y Rays 360 192 Measurements 362 1921 Procedure and Analysis 364 193 W Angular Correlation in 60Co 370 20 Experiment 12 The Compton Effect 375 201 Scattering Light from Electrons 377 2011 Relativistic Kinematics 377 2012 Classical and Quantum Mechanical Scattering 379 CONTENTS 202 Measurements 2021 Procedure 2022 Analysis 203 Advanced Topics 2031 Recoil Electron Detection 2032 Extracting the Differential Cross Section Principles of Quantum Physics A1 Photons A2 Wavelength of a Particle A3 Transitions between Bound States Principles of Statistical Mechanics B1 The Ideal Gas B2 The Maxwell Distribution Principles of Mathematics C1 Derivatives and lntegrals C2 Taylor Series C3 Natural Logarithms C4 Complex Variables D A Short Guide to MATLAB xiii 382 385 387 391 391 393 398 398 400 403 406 411 412 414 416 419 CONTENTS D1 A MATLAB Review 419 D2 Making Fancy Plots in MATLAB 424 D21 Drea7s Handle Graphics Primer 425 Chl Data Taking and Presentation Progress is made in the physical sciences through a simple process A model is developed7 and the consequences of the model are calculated These con sequences are then compared to experimental data If the consequences do not agree with the data7 then the model is wrong7 and it should be discarded After enough successful comparisons with data7 however7 a model becomes widely accepted7 and progress goes on from there Obviously7 it is crucial that the data be correct Furthermore7 the accu racy of the measurement must also be reported so that we know how strong a comparison we can make with the model Finally7 since it is likely that many people will want to compare their models to the data7 the experimental results must be reported clearly and concisely so that others can read and understand it The purpose of this chapter is to give you some ideas on how to take data correctly 7 and how to report it clearly However7 every experiment is different7 so these guidelines can only serve as a broad basis You will gain experience as you do more experiments7 learning rules for yourself as you go along We will use some loose language7 especially in this chapter Experimental Physics is a subject that can only be truly learned from experience7 and terms like settings and uncertainties will become much clearer when you7ve 2 CH 1 DATA TAKING AND PRESENTATION done your time the laboratory However we attempt to at least roughly de ne terms as we go along For starters we take the term quantity77 to be the result of some measurement like the number read off a meter stick or a voltmeter Things that you can change by hand which affect the quantity77 you want to measure are called settings I will often resort to saying something like and your intuition will get better after some experience77 I apologize but it is very hard to tell someone how to be a good experimenter You have to learn it by being shown how and then working on your own There is at least one book however which contains many good ideas about carrying out experiments 0 Practical Physics G L Squires Third Edition Cambridge University Press 1991 11 Your Log Book Keep a log book Use it to record your all your activities in the lab such as diagrams of the apparatus various settings tables of measurements and anything you may notice or realize as you go through your experiment This log book will be an invaluable reference when you return to your data at any later time and you want to make sense of what you did in the lab The book itself should have a hard binding with pages that wont get ripped out easily If you make plots on graph paper or a computer they can be attached directly into the log book with tape or staples However a good log book has pages with both vertical and horizontal rulings so that you can make hand drawn graphs directly on the page Never write data down on scratch paper so that you cari do work with it before putting it your log book Your log book should be kept neat but not too neat What7s important is that you record things so that you can go back to them at a later time and remember details of what you did Record your activities with the date and time especially when you7ve returned to recording things after a delay When you are setting up your experiment don7t worry about writing everything down as you go along but wait until things make some sense to you That 12 COMMON SENSE 3 way7 whatever you write down will make better sense when you go back to read it later Some scientists keep a log book as a daily diary7 recording not only their rneasurernents7 but lecture and seminar notes and other similar things A good tip is to leave the rst few pages blank7 and ll them in with a table of contents as you go along How you organize your lab books is up to you7 but it is probably a good idea to keep a lab book speci cally for your lab course 12 Common Sense For virtually any experirnent7 there are some good rules to keep in mind while you are taking data It is a good idea to step back once in a while7 during your experirnent7 and ask yourself if you are following these rules In later chapters we will be more precise with language regarding exper irnental uncertainty77 or measurement error For the time being7 however7 just take these terms at face value They are supposed to indicate just how precisely you have measured the desired quantity 121 Use Redundancy If you measure something with the various settings at certain values7 you should in principle get the same value again at some later time if all the settings are at their original values This should be true whether you changed the settings in between7 or if you just went out for a cup of coffee and left the apparatus alone It is always a good idea to be redundant in your data taking That is7 check to make sure you can reproduce your results In practice7 of course7 you will not get the same result when you come back to the same settings This is because any one of a number of things which you did not record like the room ternperature7 the proximity of your lab partner7 the phase of the moon will have changed and at least some 4 CH 1 DATA TAKING AND PRESENTATION of them are likely to affect your measurement in some subtle way With some experience7 you will be able to estimate what is or is not an acceptible level of reproducibility In any case7 the degree to which you can reproduce your results will serve as a measure of your experimental uncertainty for that quantity Be aware of any trends in your measurements as you take data You can be redundant also by speci cally taking data with settings that test any trends that you notice If you expect data to follow a trend based on some speci c model7 then take more data than is necessary to determine the parameters of the model For example7 suppose you are testing the notion that temperature T is a linear function of pressure P7 ie TabP Then7 the parameters a and b can in principle be determined with only two measurements of temperature at speci c settings of the pressure This is not redundant7 however7 and you should take data at more pressure settings to con rm that the linear relation is correct If it is not7 then that tells you something important about either your experimental setup or your model7 or both It is natural to take data by changing the settings monotonically That is7 to increase or decerease a setting over the range you are interested It is a good idea to at least go back and take a couple of points over again7 just to make sure things have not drifted77 while you took your data A more radical alternative is to take your data at more or less random values for the settings Don7t drive yourself crazy by changing more than one setting at a time while you are making measurements Unless you are testing some trend you may have noticed7 you will certainly want to go back to nd out each of the settings affected the measured quantity 122 Be Precise But Don t Go Overboard It is of course important to strive for as much precision as possible in your measurements However7 do not waste your time measuring one particular 12 COMMON SENSE 5 quantity very precisely ifthe result your are ultimately interested in depends on some other quantity which is known much more poorly For example suppose you want to ultimately determine the velocity 1 for some object moving in a straight line You do this by measuring the distance L that it travels in a period of time t ie 1Lt The relative precision of L and t contribute equally to the relative precision of t We will return to this in a later chapter when we discuss experimental uncertainties That is if both L and t are both known to around 10 they will both contribute to the uncertainty in 1 However there is no point in trying to gure out a way to measure It say to 1 if you cannot measure L to comparable precision Your good idea for measuring t although it may be useful and satisfying for other reasons will not help you determine 1 much more precisely It is important to keep in mind how precisely you are measuring the various quantities that go into your nal result With experience you will develop a good insight for knowing when enough is enough 123 Measure Ratios Whenever you can use your apparatus to determine ratios of quantities mea sured at different settings This is a very useful technique since common factors cancel when you take a ratio and the uncertainty in these factors cannot affect the ratio Hence a measurement of a ratio will be inherently more precise than a measurement of an absolute quantity Some of the quan tites we will measure in the experiments are ratios and they typically are determined with relatively high precision Even if the ultimate goal of the experiment cannot be expressed as a ratio try to nd ratios among your data that you can use to test the model For example suppose you want to determine the resistivity p of some metal sample from the decay lifetime 739 of some transient voltage signal Your model says that 739 depends on p through the relation TWM 6 CH 1 DATA TAKING AND PRESENTATION where R is the radius of the sample Even though you cannot determine 739 directly from a measurement of a ratio you can measure 739 for two different samples of the same metal but with different radii R The ratio of the lifetimes should be the same as the ratio of the squares of the sample radii and this is a good check on your procedure This and other examples will be pointed out along the way as we discuss the various experiments The determination of the lifetime of the free neutron is a good historical example of the triumph of ratios over absolute measurements Free neu trons decay with a half life of around 10 minutes Furthermore up until quite recently samples77 of free neutrons were only available in fast moving streams from nuclear reactors Through the 1970s the neutron lifetime was determined through two absolute measurements one of the decay rate as the stream passed through some detectors and the other of the ux of neutrons in the stream itself The various measurements of different groups did riot agree with each other and the resulting large uncertainty in the lifetime had se rious consequences in astrophysics and particle physics Then in the 1980s using a result based on the accepted model for neutron decay a different group measured a sirigle ratio which agreed with previous but less precise measurements ofthis ratio and nally pointed the way to the correct value of the neutron lifetime A ne account of these measurements is given in How Long Do Neutrons Live by SJ Freedman in Comments in Nuclear and Particle Physics 191990209 124 Avoid Personal Bias Nobody starts working on an experiment without at least a rough idea of what he or she is supposed to measure It is impossible therefore to have no notion of what to expect from the measurement of some quantity for some range of settings Sometimes though the result of a measurement is quite surprising and may be an important clue to how nature works You must walk a pretty ne line between what you expect and what you are trying to learn This will again become easier and more natural with experience Never fudge your data to give you the answer you ewpeet You will not learn anything from this and you may miss something very important There 13 TABLES AND PLOTS 7 are several great examples in the history of science where highly regarded researchers end up with egg on their faces for not keeping this in mind One example of this is documented in a very readable paper How the First Neutral Current Experiments Ended by Peter Galison in Review of Modern Physics 551983477 13 Tables and Plots A picture is worth a thousand words It is always best to display your data using either a table or a plot or both Tables are particularly useful if you want someone else to be able to take your numbers and test a different model with them Plots are best if you want to show features in your data that may be particularly important such as peaks or valleys that demonstrate some phenomenon happening at a particular setting or trends like linear or exponential behavior which may or may not support some speci c model It is a real art to know just how much information to include on a table or plot Too little data can leave the reader without enough to gure out what can be concluded from the experiment On the other hand if you put too much on the page it is very frustrating to know exactly what the important point is As with most things in Experimental Physics experience will be the most important teacher 131 Tables of Data and Results In the old days data was recorded directly from the instruments into the log books In modern times however we usually use some sort of computerized interface to gather the data In either case it is a good idea to keep the raw data as we call it in the log book Of course be judicious in what you call raw data If you read line positions from a spectrum with a thousand data points in it just record the line positions and not all the data points It is smart to always record data points exactly as you read them from the instrument instead of doing any conversions in your head or heaven 8 CH 1 DATA TAKING AND PRESENTATION forbid on scratch paper Record all conversion factors or offset values in your log book Always put labels at the head of columns or rows The labels should be terse but descriptive of the setting value or of the measured quantity7 and you should keep using that notation as you do calculations and analysis in your log book Always include the units along with the label7 and try to stick with standard conventions When recording nurnbers7 make sure you keep enough signi cant gures depending on the precision you expect to be irnportant7 but not too many Lets do an example Suppose you are measuring the time period AT of some oscillating signal using an oscilloscope as a function of some relative pressure setting PEEL measured with a vacuum gauge We will discuss various laboratory instruments in a later chapter You make a table in your log book that looks like the following PEEL AT in Hg div 275 53T 25 50 20 45 145 41 5 701 0 66 J01 rns per division 50 us per division Notice that in the middle of taking the data7 you found it was better to switch the time base of the oscilloscope You did so and noted the different conversion factors Now lets suppose that you want to do some calculations with this data so that you can test some model If you leave room in the table7 you can put the results of the calculations right there In this case7 there is not so much data and we can do this without making the table too crowded The model is best described by its dependence of the frequency V 1AT on the 13 TABLES AND PLOTS 9 internal pressure P The table might then be extended in the following way PEEL AT in Hg div PPATM V kHz 275 53T 0083 189 25 50 017 200 20 45 033 222 145 41 052 244 5 701 083 286 0 66 1 303 J01 ms per division 50 us per division This is a clear7 concise description of the data you took7 and the numbers are available to someone who may have some other idea of how to look at your data If you want to examine how well a particular model might compare to this result7 the rst thing to do is make a plot 132 Making Plots It is handy to plot the results listed in a table That makes it easy to refer back and forth between the table and the plot7 picking off details visually on the plot and reading the relevant numbers from the table For the data listed in the above table7 we7ve plotted the analyzed quantites in Fig 11 This picture could easily and quickly be made by hand7 directly in the log book Some important things can immediately be learned from this plot First7 we7ve drawn a straight line through the data points and it is clear that our results show that to a good level of accuracy7 the frequency depends linearly on pressure Note that we have plotted the data with a suppressed zero77 on the vertical axis This is a useful technique when the data covers only a limited range7 but you should be careful to make it clear when an axis does not start at zero The slope of the line can easily be read off the plot7 and its value compared to the model prediction 10 CH 1 DATA TAKING AND PRESENTATION Plotting example made on 26Feb95 I I I I Frequency kHz N N A 07 N N I 07 08 09 1 I I I 04 05 06 Pressure aim Figure 11 An Example of Plotting Data It is a good idea to choose the axes of your plot so that you can compare the behavior to a particular model You can do this easily by eye if the model predicts a straight line when you plot your data lfthe model predicts a linear dependence like in the above example then a simple plot on linear axes will do However7 if it predicts some other kind of dependence7 you have to resort to different ways of plotting the data For example7 if the model predicts an exponential dependence7 eg N Noe tT as in radioactive decay7 then it is best to plot log N log N0 7 log 67 gtlt t versus t in which case you again get an easy to see straight line dependence7 where the slope determines the value of 739 If the expected dependence is a power law function7 eg g 90V as in the gain of a photomulitplier as a function of voltage7 then logg plotted against logV gives a straight line whose intercept determines go and slope determines 71 Special graph paper or axis scaling called semilog77 or log log77 allows you to plot the quantites directly without having to take the logarithms 14 USING COMPUTERS 11 Table 11 Linear Axis Scaling in Plots Model Best Scale Slope lntercept y Ax B Linear A B y A63 Semilog B log 6 logA y AzB Log Log B logA yourself These different choices are listed in Table 11 In cases where the model is more complicated7 you can still plot the data in a way that allows you to easily see a straight line dependence For example7 if y xA 39snL B7 then plot y2 as a function of 23 In experiment 12 on Compton Scattering7 you will learn that the scattered photon energy E depends on the incident energy E and the scattering angle 6 in the following way 3 1 1 7 cost In this case7 you can plot the quantity EE as a function of 1 7 cos 9 and the result should be a straight line with slope Em and intercept at 1 The data plotted in Fig 11 simply shows data points We will later see how we determine an uncertainty77 with each of these data points7 usually associated with the quantity plotted on the vertical axis In this case7 the data points are plotted with an error bar 7 that is7 a symbol with a vertical line drawn through it The limits of the vertical line indicate the range of uncertainty77 associated with that point We will see many examples of this when we describe the experiments 14 Using Computers Nothing can replace a hand drawn plot in your log book7 as you take your data7 as a check that things are proceeding normally I urge all of you to follow this practice when you are actually running your experiment However7 a neater presentation is of course possible using any of a number of computer 12 CH 1 DATA TAKING AND PRESENTATION programs designed to tabulate and plot data A different use of computers is to actually help you analyze data7 not just plot it7 and most programs allow you to do some of both Be aware that even though different programs may all claim to be exible at some level7 they are all written with speci c priorities and audiences in mind The program you like to use will likely come down to your personal taste Following are some thumbnail sketches of programs that run either on PCs or Mac7s7 or ones that run on the Rensselaer Computer Services RCS Unix system For most examples in this book7 however7 I will use the program MATLAB which is available on PC7s7 Mac7s7 and on RCS 141 Programs for the PC There are zillions of PC plotting and analysis programs out there Some are very inexpensive and some are very pricey What they are capable of is pretty much correlated with their cost7 but that doesnt mean that you will nd more expensive programs more useful Following are some of the programs either on the PCs in the Physics Department7 or available at the ITS Product Center A more complete list and descriptions were published in the Spring 1994 Physics Courseware Communicator o The student edition of MATLAB I recommend this program More on this below GRAPH HI CRICKET GRAPH This is a simple7 easy to use7 plotting program with data entered on a spread sheet The plots are high quality and have plenty of useful options The program allows for some very basic analysis options and curve tting7 but not advanced enough for many of things you will need for this course FG SCHOLAR This is a good program for scienti c data plotting7 at a reasonable price Besides producing ne plots7 it has very sophisticated tools for data analysis including curve tting 1This quarterly publication reviews physicsrelated software7 mainly for educational user It is available from the Physics Courseware Evaluation Project PCEP at North Carolina State University Their email address is PCEP NCSUiEDUi 14 USING COMPUTERS 13 DELTAGRAPH PRO This is a higher level program than GRAPH HI and costs around twice as much The graphics are a bit more so phisticated and there are some more options for data analysis but the primary audience is not scienti c PSI PLOT This is a relatively sophisticated package aimed at scien tists and engineers A new release version 30 contains many of the analysis options you are likely to encounter in this course It is a bit expensive but we do have a copy on one of the PC7s in the student laboratory You are welcome to try it out o EXCEL Many of you are familiar with this program basically a spread sheet with graphics However it attempts to be very broad based and is therefore hard to adapt directly to the sorts of things you will need to do 142 Programs on RCS The people at lTS maintain a bunch of programs that you can use These programs are generally more sophisticated that what you get on a PC for two reasons One is that ROS has lots more memory and disk space than what you get on a PC and unless there is a lot of traf c the computers you use are a lot faster The second reason is that the University pays for the programs and they can afford some very nice packages If anything you might want to buy some documentation for the program or programs you settle on but in some cases that documentation is free and available on RCS itself You can use the Unix man pages to nd out more about these programs and where to go to get more documentation Note that you can use the SUN workstations and the MTOOLS utilities to reac PC compatible oppy disks on RCS This is a ne way to transfer data from the lab to ROS Another way is to use FTP with PC7s that are connected to the campus network 0 MATLAB I will use MATLAB for most of the examples in this course More information is in the next sections 0 GNUPLOT This is a pretty simple to use plotting program but it has CH 1 DATA TAKING AND PRESENTATION almost no analysis capabilities One very nice feature is that you can plot combinations of standard built in functions on top of your data points The program should be able to do all you need in this course7 so far as plotting is concerned XMGR The subtitle for the manual calls XMGR Graphics for ex ploratory data analysis 7 and that is pretty accurate You can actually produce wonderful looking plots7 and do rather sophisticated things with your data7 including tting and manipulations The program also works with the X11 interface so that most of your control can be window driven7 although you dont have to do things that way The biggest problem with the program is that the documentation is not easy to read7 and it will take some practice to get good at it o MAPLE Most of you are familiar with MAPLE from your math courses Recall that this program is designed for symbolic manipulation7 not data manipulation That is7 it works well with formulas7 but can be hard to use when massaging data It can be used this way7 however7 so if you7re adept at MAPLE7 you might want to use it for data analysis as well as plotting 143 MATLAB MATLAB is a numerical analysis package that is ideally suited for data anal ysis It is easy to use7 and has most of the features you will need already built in These include tting7 integration7 differentiation7 and the like The name comes from MATriX LABoratory 7 which reminds us that data is in ternally stored and manipulated as matrices We will refer to MATLAB throughout these notes7 including speci c exam ples for the various experiments General information for data analysis can also be found in Sec 6237 and various sections of Chapter 97 in particular sections 921 and 95 14 USING COMPUTERS 15 Making Plots with MATLAB MATLAB also has sophisticated plotting capability Note7 however7 that your emphasis should be on data analysis7 not making beautiful plots The plot in Fig 11 was made with MATLAB using the following commands X0083 017 033 052 083 098 y189 200 222 244 286 303 Xl0 1 y1179 306 plotxy o Xlyl Xlabel Pressure atm 7 ylabel Frequency kHz 7 title Plotting example made on date print dps plexmps Clear X y X1 yl In this example7 data is entered line by line The semicolon g after each data line is not necessary7 but if it is not included7 MATLAB echos the values of the newly created variable The plot function has a number of arguments7 and we specify the 07 which means plot the points as circles for data points7 and no option to just connect points with a straight line The axes are labeled and a title is added as shown Note that the arguments to these functions are in fact matrices of character strings7 and that is why we enclose the argument to title in square brackets The print command as used here generates a POSTSCRIPT le which can be stored or sent to your favorite printer If no options are given to the print command7 then the output automatically goes to the default printer Finally7 the data variables we de ned in the beginning are cleared7 freeing up the memory they required You can change lots ofthings on plots7 like the character size for example7 using the handle graphics capability in MATLAB Refer to Appendix D to learn the basics 16 CH 1 DATA TAKING AND PRESENTATION Entering Data into MATLAB For larger amounts of data7 you can tell MATLAB to retrieve data from a separate le7 instead of having to type all the numbers in by hand This is done with the MATLAB command load which7 for example7 will read a two column ascii data le with 71 lines into a n gtlt 2 matrix The name of the matrix is the same as the name of the le with the extension stripped off lndividual vectors of data can be extracted from this matrix For example7 if the name of the le is minedat then the MATLAB commands load minedat Xmine 1 ymine 2 create two vectors x and y each of which contains the n elements of the two columns in the le A different approach is to read77 the numbers in whatever format they were written using commands like fidfopen scllis afscanffid 7701 7 fclosefid which reads a column of numbers in the le sc1lis77 into a vector 3 The format control f should be familiar to C programmers Note that the vector is created by transposing the list read with fscanf These techniques should be particularly useful to you when reading data transferred to RCS from a oppy disk or through an ethernet connection Keeping Track of Things Anytime in the middle of a MATLAB session7 you can type the command whos to get a list of the variables you7ve created and their type The command who just gives you the list of names These can be very useful if you get confused regarding whats been created in the course of entering commands 14 USING COMPUTERS 17 Commands do not need to be entered to the command line for MATLAB Instead they can be created with some editor and stored in a le with the extension m Just entering the name of the le without the m to the MATLAB command line executes the command in this le Further Documentation on MATLAB Remember that you should use MATLAB primarily for data analysis not data plotting We will refer to the relevant commands and show examples along the way in the rest of this book There is a built in help documentation for MATLAB that should help you nd your way once you get started There are a number of other sources of MATLAB documentation 0 The MATLAB Documentation Set The MathWorks which contains several separate publications on various ways to use and modify MATLAB The ones most useful to you at the beginning would be i The MATLAB User s Guide a short description of what MATLAB can do and a tutorial introduction 7 The MATLAB Reference Guide which is a complete listing of all the standard MATLAB functions The Student Edition of MATLAB Prentice Hall 1994 which combines the Users Guide and Reference Guide from the stan dard documentaion set and can be purchased separately from the pro gram lt comes with the software package you can purchase from the ITS Product Center Numerical Methods for Physics Alejandro Garcia Prentice Hall 1994 a good book on numerical methods which uses MATLAB for most of the programming examples You might also browse the World Wide Web home page of The MathWorks at httpwwwmathworkscom which contains lots of useful information in cluding a list of books which use and refer to MATLAB 18 CH 1 DATA TAKING AND PRESENTATION 15 Formal Lab Reports When you are nished with an experiment7 or some part of it7 you may have to write a formal report on what you7ve done This is certainly the case if you want to publish your results in a scienti c journal Different people have different ideas about what these reports should look like7 and papers for journals have well de ned formats that have to be followed Here7s a hint for writing up papers or lab reports Before you start writ ing7 think about how you might explain your work to someone Better yet7 nd someone who will listen to you explain your experiment to them7 but who is riot familiar with it You7ll be surprised to see how clearly you can organize your thoughts this way The main sections of a formal report or paper are likely to include the following Title Give some thought to the title of the report It must be terse7 but still let the reader know what it is about Dont forget that titles of papers are entered in data bases used for computerized literature searches7 so try to include words that will make your paper show up in a typical search on the subject Abstract This is a concise7 self contained summary of the exper iment It should report the method7 conclusion7 and an assessment of the accuracy andor the precision of the result The abstract is a summary of the whole paper It is riot an introduction Introduction and Theory Write what you expect to learn and a general description of the experiment You should include relevant equations and formulas7 and refer to previous work on the subject 0 Experimental Setup Describe the apparatus including all relevant detail Diagrams with symbols7 standard where available7 are a good idea Refer back to these diagrams when writing the Procedure 0 Procedure and Data lndicate how you proceeded to take data Basic analyses used to process the data can be included here Tables I m EXERCISES 19 summarizing the results are a good idea Keep in mind aspects of the procedure that affect the accuracy and the precision of the data What limits the precision 0 Interpretation and Discussion This section should contain any detailed analysis on the data7 particularly where it applies to testing a certain hypothesis Discuss the result7 and whether or not it makes sense Derive whatever quanitites you can from the data7 and interpret them 0 Conclusion Summarize the experience of this measurement You may want to include suggestions for further work7 or for changes andor improvements to the apparatus 0 References List all cited references in a separate section 0 Appendices If you want to include things like raw data7 calculations7 detailed equipment descriptions7 and so on7 you should put them in Appendices They should be there if the reader wants to go into the work in more detail7 but should not be necessary for understanding the motivation or interpretation of the measurement It is important to include citations to important literature relevant to your work Tables and plots should be used wherever appropriate to make your point 16 Exercises 1 The following table lists data points for the decay rate in countssec of a radioactive source Time Rate Time Rate Time Rate sec sec sec sec sec sec 06 184 20 302 36 172 08 106 24 261 40 161 12 804 28 208 42 157 16 610 30 150 43 185 20 CH 1 DATA TAKING AND PRESENTATION Plot the data using an appropriate set of axes7 and determine over what a range of times the rate obeys the decay law R Roe tT b Estimate the value of R0 from the plot c Estimate the value of 739 from the plot d Estimate the value of the rate you expect at t 6 sec 2 An experiment determines the gravitational acceleration g by measuring the period T of a pendulum The pendulum has an adjustable length L These quantities are related as T27TZ 9 A researcher measures the following data points Data L T Point prulp klotz l 06 14 2 15 19 3 20 26 4 26 29 5 35 34 One of these data points is obviously wrong Which one Ch2 Basic Electronic Circuits Nearly every measurement made in a physics laboratory comes down to de termining a voltage It is therefore very important to have at least a basic understanding of electronic circuits before you start making physical mea surements It is not important to be able to design circuits or even to com pletely understand a circuit given to you but you do need to know enough to get some idea of how the measuring apparatus affects your result This chapter introduces the basics of elementary passive electronic cir cuits You should be familiar with the concepts of electric voltage and cur rent before you begin but something on the level of an introductory physics course should be suf cient It is helpful to have already learned something about resistors capacitors and inductors as well but you should get what you need to know about such things out of this chapter at least as far as this course is concerned There is a very little bit at the end about diodes and transistors but there is more on them in the experiments in which they are used This chapter is not a substitute for a course in electronics design There are of course lots of books on the subject and you should get one that you are comfortable with Solid state electronics is an ever growing eld so dont get hooked on a very old book An excellent up to date text and reference book on electronics that most people in the business use or at least have a copy of is 21 22 CH 2 BASIC ELECTRONIC CIRCUITS o The Art of Electronics by Paul Horowitz and Win eld Hill Second Edition Cambridge University Press 1989 A student manual for this book is also available Another nice book which includes a few introductory chapters on solid state electronics including the physics behind diodes and transistors is o Emperirnental Physics Modern Methods by R A Dunlap Oxford University Press 1988 A good introduction to the basics of electric circuits is found in 0 Physics Robert Resnick David Halliday and Kenneth Krane John Wiley and Sons Fourth Edition 1992 Chap31 Capacitors and Dielectrics Chap32 Current and Resistance Chap33 DC Circuits Chap38 Inductance Chap39 AC Circuits 21 Voltage Resistance and Current Lets start at the beginning Figure 21a shows the run of the mill DC current loop It is just a battery that provides the electromotive force V which drives a current i through the resistor B This is a cumbersome way to write things however so right off the bat we will use the shorthand shown in Fig 21b All that ever matters is the relative voltage between two points so we specify everything relative to the common or ground There is no need to connect the circuit loop with a line it is understood that the current will ow from the common point up to the terminals of the battery 21 VOLTAGE RESISTANCE AND CURRENT V ER L a b Figure 21 The simple current loop a in all its glory and b in shorthand 211 Loop and Junction Rules The concept of electric potential is based on the idea of electric potential energy7 and energy is conserved This means that the total change in electric potential going around the loop in Fig 21a must be zero In terms of Fig 21b7 the voltage drop77 across the resistor R must equal V lt7s actually a pretty trivial statement when you look at it that way This is a good time to remind you of the de nition of resistance7 namely R is just the voltage drop across the resistor divided by the current through the resistor In other words7 the voltage drop through a resistor R is equal to ZR where 239 is the current through it In terms of the simple loop in Fig 217 V ZR The Sl unit of resistance is VoltsAmps7 also known as the Ohm 9 Just about all the resistors you will care about in this course obey Ohm s Law7 which just states that the resistance R is independent of the current 239 ln fact7 we nearly always use the symbol R to mean a constant value of a resistance7 that is7 a resistor that obeys Ohm7s Law 24 CH 2 BASIC ELECTRONIC CIRCUITS Electric current is just the ow of electric charge 2 E dqdt to be precise and electric charge is conserved This means that when there is a junction77 in a circuit like the one shown in Fig 22 the sum of the currents owing into the junction must equal the sum of the currents owing out In the case of Fig 22 this rule just implies that 21 22 23 It doesnt matter whether you specify the current owing in or out so long as you are consistent with this rule Remember that current can be negative as well as positive Figure 22 A simple three wire circuit junction These rules and de nitions allow us to determine the resistance when re sistors are connected in series as in Fig 23a or in parallel as in Fig 23b In either case the voltage drop across the pair must be 2R where 2 is the current through set For two resistors R1 and R2 connected in series the current is the same through both so the voltage drops across them are 2R1 and 2R2 respectively Since the voltage drop across the pair must equal the sum of the voltage drops then 2R 2R1 2R2 or R R1 R2 Resistors in Series lf R1 and R2 are connected in parallel then the voltage drop across each are the same but the current through them is different Therefore 2R 21R1 22R2 Since 2 21 22 we have 1 1 1 E P E Resistors in Parallel Remember that whenever a resistor is present in a circuit it may as well be some combination of resistors that give the right value of resistance 22 CAPACITORS AND AC CIRCUITS 25 VVV NVV 3A1 R1 R2 R 2 a b Figure 23 Resistors connected a in series and b in parallel 212 The Voltage Divider A very simple7 and very useful7 con guration of resistors is shown in Fig 24 This is called a voltage divider because of the simple relationship between the voltages labeled VOUT and VIN Clearly VIN 239 R1 R2 and VOUT MRZ where 239 is the current through the resistor string Therefore R2 VOUT VINi 21 R1 R2 That is7 this simple circuit divides the input voltage into a fraction deter mined by the relative resistor values We will see lots of examples of this sort of thing in the laboratory Don7t let yourself get confused by the way circuits are drawn lt doesn7t matter which directions lines go in Just remember that a line means that all points along it are at the same potential For example7 it is common to draw a voltage divider as shown in Fig 25 This way of looking at it is in fact an easier way to think about an input voltage and an output voltage 22 Capacitors and AC Circuits A capacitor stores charge7 but does not allow the charge carriers ie elec trons to pass through it It is simplest to visualize a capacitor as a pair 26 CH 2 BASIC ELECTRONIC CIRCUITS V 111 T R 1 out ER 2 Figure 24 The basic voltage divider of conducting plates7 parallel to each other and separated only by a small amount Sorne capacitors called parallel plate capacitors are actually constructed this way7 but the kind used in circuits are usually little cerarnic disks with a bulge in the middle and two wire leads sticking out If a capacitor has a potential difference V across its leads and has stored a charge q on either side7 then we de ne the capacitance C E qV It is easy to show for a parallel plate capacitor C is a constant value independent of the voltage It is not so easy to do this in general7 but it is still true for the most part The Sl unit of capacitance is VoltsArnperes7 also known as the Farad As it turns out7 one Farad is an enormous capacitance7 and laboratory capacitors typically have values between a few rnicrofarads MF down to a few hundred rnicrornicrofarads MMF or picofarads People who work with circuits a lot are likely to refer to a picrofarad as a pu It is pretty easy to gure out what the effective capacitance is if capacitors are connected in series and in parallel7 just using the above de nitions and 22 CAPACITORS AND AC CIRCUITS 27 V C VVV T o m Vout R2 R13 Figure 25 An alternate way to draw a voltage divider the rule about the total voltage drop The answers are 1 1 1 5 a 62 Capacitors in Series and C C1 C2 Capacitors in Parallel That is7 just the opposite from resistors Now lets think about what a capacitor does in a circuit Lets take the resistor R2 in the voltage divider of Fig 24 and replace it with a capacitor C This is pictured in Fig 26 The capacitor does not allow any charge carriers to pass through it7 so the current 239 0 Therefore the voltage drop across the resistor R is zero7 and VOUT7 the voltage across the capacitor C7 just equals VIN What good is this We might have just as well connected the output terminal to the input To appreciate the importance of capacitors in circuits7 we have to consider voltages that change with time 221 DC and AC circuits If the voltage changes with tirne7 we refer to the system as an AC circuit If the voltage is constant7 we call it a DC circuit AC rneans alternating current77 and DC rneans direct current These names are old and not very descriptive7 but everyone uses them so we are stuck with them 28 CH 2 BASIC ELECTRONIC CIRCUITS V 1 R T E T Figure 26 A voltage divider with a capacitor in it Lets go back to the voltage divider with a capacitor7 pictured in Fig 267 and let the input voltage change with time in a very simple way That is7 take lNt 0 fort 0 V fortgt0 A coio VV and assume that there is no charge q on the capacitor at t 0 Then for t gt 07 the charge qt produces a voltage drop VOUTt qtC across the capacitor The current z t dqdt through the divider string also gives a voltage drop ZR across the resistor7 and the sum of the two voltage drops must equal V In other words d dV q VOUT RCioUT VVOUTZRVOUTRdt7 dt 24 and VOUT0 0 This differential equation has a simple solution It is V0UTt V 1 GitRC 25 22 CAPACITORS AND AC CIRCUITS 29 Now it should be clear what is going on As soon as the input voltage is switched on7 current ows through the resistor and the charge carriers pile up on the input side ofthe capacitor There is induced charge on the output side of the capacitor7 and that is what completes the circuit to ground However7 as the capacitor charges up7 it gets harder and harder to put more charge on it7 and as t a 007 the current doesn7t ow anymore and VOUT a V This is just the DC case7 where this circuit is not interesting anymore The value RC is called the capacitive time constant77 and it is the only time scale we have in this circuit That is7 statements like t a 077 and t a 0077 actually mean t lt RC and t gt R0 The behavior of the circuit will always depend on the time relative to RC So now we see what is interesting about capacitors They are sensitive to currents that are changing with time in a way that is quite different from resistors That is a very useful property that we will study some more7 and use in lots of experiments The time dependence of any function can always be expressed in terms of sine and cosine functions using a Fourier transform It is therefore common to work with sinusoidally varying functions for voltage and so forth7 just realizing that we can add them up with the right coef cients to get what ever time dependence we want in the end It is very convenient to use the notation 1 Vt mm 26 for time varying ie AC voltages7 where it is understood that the voltage we measure in the laboratory is just the real part of this function The angular frequency w 27w where V is the frequency7 that is7 the number of oscillations per second This expression for Vt is easy to differentiate and integrate when solving equations It is also a neat way to keep track of all the phase changes signals undergo when the pass through capacitors and other reactive77 components You7ll see and appreciate this better as we go along 1lf youlre not familiar with complex numbers7 see Appendix Ci4i 30 CH 2 BASIC ELECTRONIC CIRCUITS 222 Impedance Now is a convenient time to de ne impedance This is just a generalization of resistance for AC circuits lmpedance usually denoted by Z is a usually complex quantity and usually a function of the angular frequency w It is de ned as the ratio of voltage drop across a component to the current through it andjust as for resistance the SI unit is the Ohm For linear components of which resistors and capacitors are common examples the impedance is not a function of the amplitudes of the voltage or current signals Given this de nition of impedance the rules for the equivalent impedance is the same as for resistance That is for components in series add the impedances while if they are in parallel add their reciprocals The impedance of a resistor is trivial It is just the resistance R In this case the voltage drop across the resistor in phase with the current through it since Z R is a purely real quantity The impedance is also independent of frequency in this case Things get to be more fun with capacitors In this case the voltage drop V ljem qC and the current 239 dqdt sz gtlt V661 Therefore the impedance is Vw t 1 z wt T sz Now the behavior of capacitors is clear At frequencies low compared to 1RC ie the DC limit the impedance of the capacitor goes to in nity Here the value of R is the equivalent resistance in series with the capacitor It does not allow current to pass through it However as the frequency gets much larger than 1RC the impedance goes to zero and the capacitor acts like a short since current passes through it as if it were not there You can learn a lot about the behavior of capacitors in circuits just by keeping this in mind 2M 27 There is an important lesson here Between any two conductors there is always some capacitance Therefore no matter how well some circuit is de signed there will always be some stray capacitance around however small Consequently the circuit will always fail above some frequency because effec tive shorts appear throughout You can only keep the stray capacitance so small especially in integrated circuit chips where things are packed tightly 22 CAPACITORS AND AC CIRCUITS 31 together7 and this is a practical limitation for all circuit designers 223 The Generalized Voltage Divider We can easily generalize our concept of the voltage divider to include AC circuits and reactive ie frequency dependent components like capacitors We will learn about another reactive cornponent7 the inductor7 shortly The generalized voltage divider is shown in Fig 27 In this case7 we have Figure 27 The generalized voltage divider VOUT W71 VIN W71 VIN W71 95m 28 Z2 Z1 Zz where we take the liberty of writing the impedance ratio ZlZl Z27 a complex nurnber7 in terms of two real numbers 9 and b We refer to g lVOUTlllINl as the gain of the circuit7 and b is the phase shift of the output signal relative to the input signal For the simple resistive voltage divider shown in Fig 24 and Fig 257 we have 9 RlRl R2 and b 0 32 CH 2 BASIC ELECTRONIC CIRCUITS I I m 39le DE IVOUTI n5 9Vw DA DZ m 3 5 9 lt02 lt04 lt03 lt08 1 2150 l I WEI ZEI 3E AU SD EU 70 TlME Figure 28 Input and output voltages for the generalized voltage divider That is7 the output signal is in phase with the input signal7 and the amplitude is just reduced by the relative resistor values This holds at all frequencies7 including DC The relative phase is an important quantity7 so lets take a moment to look at it a little more physically If we write VIN V661 then according to Eq 28 we can write VOUT gljelw Since the measured voltage is just the real part of these complex expressions7 we have VIN Vb coswt VOUT 9V6 COSQUI 15 These functions are plotted together in Fig 28 The output voltage crests at a time different than the input voltage7 and this time is proportional to the phase To be exact7 relative to the time at which VIN is a maximum7 i i i 45 i 45 Time of maX1mum VOUT 7 if gtlt T i i 27139 u where T 27rw is the period of the voltage uctuations This time lag can make all the difference in the world in many circuits Now lets consider the voltage divider in Fig 26 Using Eq 28 we nd le 1 V V 71 OUT INRF10 IN1szC 23 INDUCTORS 33 The gain 9 of this voltage divider is just 1 w2R202712 and you can see that for w 0 ie DC operation the gain is unity For very large frequencies though the gain goes to zero The gain changes from unity to zero for frequencies in the neighborhood of 1RC We have said all this before but in a less general language However our new language tells us something new and important about VOUT namely the phase relative to VIN Using equations 02 and 03 we nd that 1 1 7 szC 1 7 Z5 1 w2R20212 1szO 1w2R202 In other words the output voltage is phase shifted relative to the input voltage by an amount b 7 tan 1wRC For in 0 there is no phase shift as you should expect but at very high frequencies the phase is shifted by 23 Inductors Just as a capacitor stores energy in an electric eld an inductor stores energy in a magnetic eld An inductor is essentially a wire wound into the shape of a solenoid The symbol for an inductor is At rst you might think A wire is a wire so what difference could it make to a circuit7 They key is in the magnetic eld that is set up inside the coil and what happens when the current changes So just as with a capacitor inductors are important when the voltage and current change with time and the response depends on the frequency The inductance L of a circuit element is de ned to be 7 Nlt1gt 239 L where N is the number of turns in the solenoid and ltIgt is the magnetic ux in the solenoid generated by the current 239 The 81 unit of inductance is the TeslarnZArnpere or the Henry 34 CH 2 BASIC ELECTRONIC CIRCUITS Now if the current 239 through the inductor coil is changing then the mag netic ux is changing and this sets up a voltage in the coil that resists the change in the current The magnitude of this voltage drop is dNltIgt dz39 Li V dt dt If we write V Z39Z where Z is the impedance ofthe inductor and V Vbem then V LZzwV or Z ML 29 We can use this impedance to calculate for example VOUT for the generalized voltage divider of Fig 27 if one or more of the components is an inductor You can now see that the inductor is to large extent the opposite of a capacitor The inductor behaves as a short that is just the wire it is at low frequencies whereas a capacitor is open in the DC limit On the other hand an inductor behaves as if the wire were cut an open circuit at high frequencies but the capacitor is a short in this limit You can make inter esting and useful circuits by combining inductors and capacitors in different combinations One particularly interesting combination is the series LCR circuit com bining one of each in series The impedance of such a string displays the phenomenon of resonance That is in complete analogy with mechanical resonance the voltage drop across one of the elements is a maximum for a certain value of w Also as the frequency passes through this value the relative phase of the output voltages passes through 90 If the resistance R is very small then the output voltage can be enormous in principle 24 Diodes and Transistors Resistors capacitors and inductors are linear77 devices That is we write V Z39Z where Z is some complex number which may be a function of frequency The point is though that if you increase V by some factor then you increase 239 by the same factor 24 DIODES AND TRANSISTORS 35 Diodes and transistors are examples of nonlinear77 devices Instead of talking about some irnedance Z we instead consider the relationship between V and239 as some nonlinear function What7s more a transistor is an active77 device unlike resistors capacitors inductors and diodes which are passive That is a transistor takes in power from some voltage or current source and gives an output that combines that input power with the signal input to get a response As you might guess transistors are very popular signal ampli ers although they have lots of other uses as well Instead of covering the world of nonlinear devices at this time we willjust discuss some of their very basic properties We will describe their operation in some more detail when we use them in speci c experirnents since they can be used in a large variety of ways You might know that in the old days many of these functions were possi ble with vacuum tubes of various kinds These have been almost completely replaced by solid state devices based on semiconductors 241 Diodes The symbol for a diode is where the arrow shows the nominal direction of current ow An ideal diode conducts in one direction only That is its V 7239 curve would give zero current 2 for V lt 0 and in nite 2 for V gt 0 Of course in practice the current 2 is limited by some resistor in series with the diode This is shown in Fig 29a A real diode however has a more complicated curve as shown in Fig 29b The current 2 changes approxirnately exponentially with V and becomes very large for voltages above some forward voltage drop VF For most cases a good approximation is that the current is zero for V lt VF and unlimited for V gt VF Typical values of VF are between 05 V and 08 V 36 CH 2 BASIC ELECTRONIC CIRCUITS Figure 29 Current 239 versus voltage V for a the ideal diode and b a real diode Diodes are pnjunctions These are the simplest solid state devices made of a semiconductor usually silicon The electrons in a semiconductor ll an energy band and normally cannot move through the bulk material so the semiconductor is really an insulator lf electrons make it into the next energy band which is normally empty then they can conduct electricity This can happen if for example electrons are thermally excited across the energy gap between the bands For silicon the band gap is 11 eV but the mean thermal energy of electrons at room temperature is N kT 140 eV Therefore silicon is essentially an insulator under normal conditions and not particularly useful That7s where the p and 7 come in By adding a small amount around 10 parts per million of speci c impurities lots of current carriers can be added to the material These impurities called dopants can precisely control how current is carried in the semiconductor Some dopants like arsenic give electrons as carriers and the doped semiconductor is called n type since the carriers are negative Other dopants like boron bind up extra electrons and current is carried by holes created in the otherwise lled band These holes act like positive charge carriers so we call the semiconductor p type In either case the conductivity increases by a factor of 1000 at room temperature this makes some nifty things possible 24 DIODES AND TRANSISTORS 37 So now back to the diode7 or pnjunction This is a piece of silicon7 doped p type on one side and n type on the other Electrons can only ow from p to 71 That is7 a current is carried only in one direction A detailed analysis gives the 239 7 V curve shown in Fig 29b See Dunlap for more details If you put voltage across the diode in the direction opposite to the direc tion of possible current flow7 that is called a reverse bias If you put too much of a reverse bias on the diode7 ie V lt iV lIAX it will break down and start to conduct This is also shown in Fig 29b Typical values of VIQIAX are 100 V or less 7 242 Transistors Transistors are considerably more complicated than diodes27 and we will only scratch the surface here The following summary closely follows the introduction to transistors in The Art of Electronics For details on the underlying theory7 see Dunlap A transistor has three terminals7 called the collector7 base7 and emitter There are two main types of transistors7 namely npn and 1071p7 and their sym bols are shown in Fig 210 The names are based on the dopants used in the semiconductor materials The properties of a transistor may be summarized in the following simple rules for npn transistors For pnp transistors7 just reverse all the polarities 1 The collector must be more positive than the emitter 2 The base emitter and base collector circuits behave like diodes Nor mally the base emitter diode is conducting and the base collector diode is reverse biased C40 Any given transistor has maximum values of 2390 2393 and VCE that cannot be exceeded without ruining the transistor If you are using an transistor in the design of some circuit7 check the speci cations to see what these limitng values are 2The invention of the transistor was worth a Nobel Prize in Physics in 1956 38 CH 2 BASIC ELECTRONIC CIRCUITS Collector C Base B Emitter npn pnp m Figure 210 Symbols for npn and pnp transistors 4 When rules 1 3 are obeyed7 2390 is roughly proportional to 23 and can be written as 2390 hpEz39B The parameter hpE7 also called 6 is typically around 1007 but it varies a lot among a sample of nominally identical transistors Obviously7 rule 4 is what gives a transistor its punch It means that a transistor can amplify77 some input signal It can also do a lot of other things7 and we will see them in action later on 25 Exercises 1 Consider the following simple circuit 2 5 EXERCISES 39 Let the input voltage VIN be a sinusoidally varying function with ampli tude Vb and angular frequency w a Calculate the gain 9 and phase shift b for the output voltage relative to the input voltage b Plot 9 and b as a function of wwo where we 1RC For each of these functions7 use the combination of linear or logarithrnic axes for g and for b that you think are most appropriate 2 Consider the following simple circuit vi R v n out Let the input voltage VIN be a sinusoidally varying function with ampli tude Vb and angular frequency w a Calculate the gain 9 and phase shift b for the output voltage relative to the input voltage b Plot 9 and b as a function of wwo where we RL For each of these functions7 use the combination of linear or logarithrnic axes for g and for b that you think are most appropriate 3 Consider the following not so sirnple circuit 40 a F7 0 CH 2 BASIC ELECTRONIC CIRCUITS V R V in out What is the gain 9 for very low frequencies w What is the gain for very high frequencies Remember that capacitors act like dead shorts and open circuits at high and low frequencies respectively and inductors behave in just the opposite way At what frequency do you suppose the gain of this circuit is maximized Use your intuition and perhaps some of Chapter 38 in Resnick Halli day and Krane Using the rules for impedance and the generalized voltage divider de termine the gain 9a for this circuit and show that your answers to a and b are correct 4 Suppose that you wish to detect a rapidly varying voltage signal However the signal is superimposed on a large DC voltage level that would damage your voltmeter if it were in contact with it You would like to build a simple passive circuit that allows only the high frequency signal to pass through a F7 Sketch a circuit using only a resistor R and a capacitor C that would do the job for you Indicate the points at which you measure the input and output voltage Show that the magnitude of the output voltage equals the magnitude of the input voltage multiplied by 1 1 1 u RzC2 where w is the angular frequency of the signal You may use the expression for capacitor impedance we derived in class 2 5 EXERCISES 41 c Suppose that R1kQ and the signal frequency is 1MHZ106SSC Sug gest a value for the capacitor C 42 CH 2 BASIC ELECTRONIC CIRCUITS Ch3 Common Laboratory Equipment There are lots of different kinds of laboratory equipment In fact there are too many to cover in any detail and you will learn about speci c pieces of equipment as you do the experiments However there are certain kinds of equipment common to nearly all experiments and we will talk about these in this chapter As you might imagine all of this equipment is related to generating or measuring voltage I dont know of any book that covers the speci c sorts of things in this chapter If you are interested in some speci c piece of equipment however a good place to check is with the manufacturer or distributer of a product line You can typically get good documentation for free and not always in the form of the company7s catalog 31 Wire and Cable Connections between components are made with wires We tend to neglect the importance of choosing the right wire for the job but in some cases it can make a big difference 43 44 OH 3 COMMON LABORATORY EQUIPMENT The simplest wire is just a strand of some conductor most often a metal like copper or aluminum Usually the wire is coated with an insulator so that it will not short out to its surroundings or to another part of the wire itself If the wire is supposed to carry some small signal then it will likely need to be shielded that is covered with another conductor outside the insulator so that the external environment doesn7t add noise somehow One popular type of shielded wire is the coaxial cable which is also used to propagate pulses 311 Basic Considerations Don7t forget about Ohm7s law when choosing the proper wire That is the voltage drop across a section of wire is still V ZR and you want this voltage drop to be small compared to the real voltages involved The resistance R p gtlt LA where L is the length of the wire A is its cross sectional area and p is the resistivity of the metal Therefore to get the smallest possible R you keep the length L as short as practical get a wire with the largest practical A1 and choose a conductor with small resistivity Copper is the usual choice because it has low resistivity p 169 gtlt 10 8 9cm and is easy to form into wire of various thicknesses and shapes Other common choices are aluminum p 275 gtlt 10 8 9cm which can be signi cantly cheaper in large quantities or silver p 162 gtlt 10 8 9cm which is a slightly better conductor although not usually worth the increased expense The resistivity increases with temperature and this can lead to a partic ularly insidious failure if the wire has to carry a large current The power dissipated in the wire is P HR and this tends to heat it up If there is not enough cooling by convection or other means then R will increase and the wire will get hotter and hotter until it does serious damage This is most common in wires used to wind magnets but can show up in other high power applications A common solution is to use very low gage ie very thick wire that has a hollow channel in the middle through which water ows The water acts as a coolant to keep the wire from getting too hot 1Wire diameter is usually specified by the gage number i The smaller the wire gage the thicker the wire and the larger the cross sectional area 31 WIRE AND CABLE A5 The wue rnsu1etor rnust elso wrthstend the ternpereture rnoreese end whatever else the outside environment wants to throw at It H may be oessery tor exenrple to rrnrnerse pert or e orrourt rn hqurd nrtrogen end you don t went the rnsu1etor o creek epert It should not be herd to nd e oonduotor end rnsu1etor oornhrnetron thet wru surt your purpose 312 Coaxial Cable ooexrel oeble rs e shrelded wrre The neme comes from the xi thet the ngure 31 Cutewey yrew or ooexrel oehle very srneu end ere hkeh to be obscured by some sort or eleotronro norse rn the room The outsrde oonduotor oeued the shreld makes rt drmoult ror externel electromegnetro elds to penetrete to the wrre end mrnrrnrzes the horse Thrs ousrde oonduotor rs usuelh connected to yound nd end yer rrnportent use or ooexrel oehle rs tor pulse trens htt1e drstortron trorn drspersron Short pulses oen be very c reboretory rn such epphoetrons es dxgtal slglal trensrnrssron hon detectors You have to be awexe of the chexexlenshc Imp the oehle when you use rt rn thrs wey e end rn mixer edenoequot or Coexrel oeble hes e ohereeterrstro rrnpedenoe beoeuse rt trensrnm the 46 OH 3 COMMON LABORATORY EQUIPMENT signal as a train of electric and magnetic uctuations and the cable itself has characteristic capacitance and inductance The capacitance and inductance of a cylindrical geometry like this are typically solved in elementary physics texts on electricity and magnetism The solutions are O T1xlam Lim 9x6 lnba 27139 a where a and b are the radii of the wire and shield respectively 6 and a are the permittivity and permeability of the dielectric and l is the length of the cable It is very interesting to derive and solve the equations that determine pulse propagation in a coaxial cable but we wont do that here One thing you learn however is that the impedance seen by the pulse which is dominated by high frequencies is very nearly real and independent of frequency and equal to L 1 a b Zciliilil 7 31 O 27139 E n a This characteristic impedance77 is always in a limited range typically 509 S Z0 3 2009 owing to natural values of E and a and to the slow variation of the logarithm You have to be careful when making connections with coaxial cable so that the characteristic impedance Zc of the cable is matched77 to the load impedance ZL The transmissions equations are used to show that the re ection coef cien 77 P de ned as the ratio of the current re ected from the end of the cable to the current incident on the end is given by iaiz TZLZC That is if a pulse is transmitted along a cable and the end of the cable is not connected to anything ZL 00 then P 1 and the pulse is immediately re ected back On the other hand if the end shorts the conductor to the shield ZL 0 then P 71 and the pulse is inverted and then sent back The ideal case is when the load has the same impedance as the cable In this case there is no loss at the end of the cable and the full signal is transmitted through You should take care in the lab to use cable and electronics that have matched impedances Common impedance standards are 509 and 909 31 WIRE AND CABLE 47 313 Connections Of course you will need to connect your wire to the apparatus somehow and this is done in a wide variety of ways For permanent connections especially inside electronic devices solder is usually the preferred solution You wont typically make solder joints in the undergraduate laboratory unless you are building up some piece of apparatus It is harder than you might think to make a good solder joint and if you are going to do some of this you should have someone show you who has a decent amount of experience Another type of permanent connection called crimping squeezes the conductors together using a special tool that ensures a good contact that does not release This is particularly useful if you can7t apply the type of heat necessary to make a good solder joint Again you are unlikely to encounter this in the undergraduate laboratory Less permanent connections can be made using terminal screws or binding posts These work by taking a piece of wire and inserting it between two surfaces which are then forced together by tightening a screw You may need to twist the end of the wire into a hook or loop to do this best or you may use wire with some sort of attachment that has been soldered or crimped on the end If you keep tightening or untightening screws especially onto wires with hand made hooks or loops then the wire is likely to break at some point Therefore for temporary connections it is best to use alligator clips or ba nana plugs or something similar Again you will usually use wires with this kind of connector previously soldered or crimped on the end Coaxial cable connections are made with one of several special types of connectors Probably most common is the Bayonet N Connector or BNC standard including male cable end connectors female device connectors and union and T connectors for joining cables In this system a pin is soldered or crimped to the inner conductor of the cable and the shield is connected to an outer metal holder Connections are made by twisting the holder over the mating connector with the pin inserting itself on the inner part Another common connector standard called Safe High Voltage77 or SHV works similarly to BNC but is designed for use with high DC voltages by making it dif cult to contact the central pin unless you attach it to the 48 OH 3 COMMON LABORATORY EQUIPMENT correct mate For low level measurement you must be aware of the thermal electric potential difference between two dissimilar conductors at different temper atures These thermoelectric coef cients are typically around 1 MV C but between Copper and Copper Oxide which can easily happen if a wire or terminal has been left out and is oxidized it is around 1 mV C 32 DC Power Supplies A lot of laboratory equipment needs to be powered in one way or another Unlike the typical 100 V 60 HZ AC line you get out ofthe wall socket though this equipment usually requires some constant DC level to operate One way to get this constant DC level is to use a battery but if the equipment draws much current the battery will die quickly More often we use DC power supplies to get this kind of constant DC level The power supply in turn gets its power from the wall socket Power supplies come in lots of shapes sizes and varieties but there are two general classes These are voltage supplies or current supplies and the difference is based on how the output is regulated Since the inner work ings of the power supply has some effective resistance when the power supply has to give some current there will be a voltage drop across that resistance and that will affect how the power supply works In a voltage regulated supply the circuitry is designed to keep the output voltage constant to within some tolerance regardless of how much current is drawn Typically there will be some maximum current at which the regulation starts to fail That is there is a maximum power that can be supplied Most electronic devices and detector systems prefer to have a speci c voltage they can count on so they are usually connected to voltage regulated supplies A current regulated supply is completely analagous but here the cir cuitry is designed to give a constant output current in the face of some load on the supply Such supplies are most often used to power magnets since the magnetic eld only cares about how much current ows through the coils This is in fact quite important for precise magnetic elds since the coils tend 3 3 WAVEFORM GENERATORS 49 to get hot and change their resistance In this case7 V ZR and R is chang ing with time7 so the power supply has to know to keepz constant by varying V accordingly In many cases7 a simple modi cation usually done without opening up the box can convert a power supply from voltage regulation to current regulation The output terminals on most power supplies are oating That is7 they are not tied to any external potential7 in particular to ground One output sometimes colored in red is positive with respect to the other black You will usually connect one of the outputs to some external point at known potential7 like a common ground You should be aware of some numbers The size and price of a power supply depends largely on how much power it can supply If it provides a voltage V while sourcing a current 239 then the power output is P Z39V A very common supply you will nd around them lab will put out several volts and a couple of amps7 so something like 10 W or so Depending on things like control knobs and settings to computer interfacing7 they can cost anywhere from 50 up to a few hundred So called high voltage power supplies will give several hundred up to several thousand or more volts7 and can source anywhere from a few MA up to 100 mA7 and keep the voltage constant to a level of better than 100 mV Still7 the power output of such devices is not enormously high7 typically under a few hundred watts The cost will run into thousands of dollars Magnet power supplies7 though7 may be asked to run something like 50 A through a coil that has a resistance of7 say7 2 9 In this case7 the output power is 5 kW7 and that is a force to be reckoned with Realize7 of course7 that these are all round numbers just to give you some idea of what you7ll see around the laboratory 33 Waveform Generators You might think there are things called AC power supplies7 analagous to the DC supplies we7ve just discussed Well there are7 but we don7t call them that because in general and certainly for the equipment you will see in this course they don7t supply much power lnstead7 we talk about Waveform Generators which produce an output voltage signal that varies in time 50 OH 3 COMMON LABORATORY EQUIPMENT Later in Experiment 5 we will combine a waveform generator with a DC power supply to make an AC power supply and we will talk more about that then The function Vt can be anything from a simple sine wave to an arbitrary function you program into the device but increased exibility can cost a lot of money Most waveform generators though do have at least sine waves square waves or triangle waves and can vary the frequency over a wide range Low frequencies are pretty easy to get but for very high frequencies above a MHZ or so things get much harder because of stray capacitance giving effective shorts See section 221 You can also vary the voltage amplitude and offset over several volts Sometimes instead of wanting a wave output you need a pulse That is a signal that is high for some short period of time with another coming after a much longer time Most waveform generators can accomodate your wishes either by providing an explicit pulse output or by allowing you to change the symmetry of the waveform so that the 0 to 7T portion of the wave is stretched or compressed relative to the 7T to 27f portion 34 Meters So now that you know how to get some voltage including time varying ones and how to connect these voltages using wire and cable you have to think about how to measure the voltage you create The simplest way to do this is with a meter particularly if the voltage is DC Most meters do provide you with AC capability but we won7t go into the details here An excellent reference on the subject of meters is given in the Low Level Measurements Handbook published by Keithley Instruments Inc If you want a copy call them at 216248 0400 and they will probably send you one for free As you might imagine Keithley sells meters In the old days people would use either voltmeters ammeters or ohm meters to measure voltage current or resistance respectively These days although you still might want to buy one of these specialized instruments to get down to very low levels most measurements are done with Digital Mul 35 OSCILLOSCOPES 51 timeters 7 or DMM7s for short ln fact7 some DMM7s are available now that effectively take the place of the most sensitive specialized meters Voltage and resistance measurements are made by connecting the meter in parallel to the portion of the circuit you7re interested in To measure current7 you have to put the meter in series Realize that DMM7s work by averaging the voltage measurement over some period of time7 and then displaying the result This means that if the voltage is uctuating on some time scale7 these uctuations will not be observed if the averaging time is greater than the typical period of the uctuations Of course the shorter the averaging time a meter has the higher the bandwidth77 it has7 the fancier it is and the more it costs Most of the applications in this course do not involve very low level mea surements7 but you should be aware of a simple fact just the same Meters have some effective input impedance7 so they will at some level change the voltage you are trying to measure For this reason7 voltmeters and ohm meters are designed to have very large input impedances many M9 to as high as several G97 while ammeters shunt77 the current through a very low resistance and turn the job into measuring the perhaps very low voltage drop across that resistor 35 Oscilloscopes An oscilloscope measures and displays voltage as a function of time That is7 it plots for you the quantity Vt on a cathode ray tube CRT screen as it comes in This is a very useful thing7 and you will use oscilloscopes in nearly all the experiments you do in this course A good reference is The XYZ7s of Oscilloscopes 7 published by Tektronix7 lnc7 probably the worlds largest manufacturer of oscilloscopes The simple block diagram shown in Fig32 explains how an oscilloscope works The voltage you want to measure serves two purposes First7 after being ampli ed7 it is applied to the vertical de ection plates ofthe CRT This means that the vertical position of the trace on the CRT linearly corresponds to the input voltage7 which is just what you want The vertical scale on the 52 CH 3 COMMON LAB ORATORY EQUIPMENT mute s 2 Block Dugam of an Oscilloscope CRT has e W pauem that lets ycu know wbeb Lhe mpub vnlbage 1s 351 Sweep and D igger The benzenm pmlhnn of me Luce 1s canknlled by e sweep genemas whnse eed can comm Howevez umpenuvesxynl shapesysu mm Lhe signal m sum eb me same ume 5m eve sweep and ms 1s detennmed by gelquot system The place 7 ch screen whale me bme suns 1s canknlled by e quothorizontal yoshcnquot knob on me mm panel e 11 age 1s m Just beve me soc e sweep eb Lhe lme 16 50 Hz fzequency ome aL Hegel Theze 1s usualbl e 11g on Lhe mm panel that ash scape 1s Lngezed 3 5 OSCILLOSCOPES 53 Oscilloscopes almost always have at least two input channels7 and it is possible to trigger on one channel and look at the other This can be very useful for studying coincident signals or for measuring the relative phase of two waveforms In any case7 the trigger mode can either be normal 7 in which case there is a sweep only if the trigger condition is met7 or auto where the scope will trigger itself if the trigger condition is not met in some period of time Auto mode is particularly useful if you are searching for some weak signal and dont want the trace to keep disappearing on you 352 Input Voltage Control You have several controls on how the input voltage is handled A verti cal position knob on the front panel controls where the trace appears on the screen You will nd one of these for each input channel The input coupling can be set to either AC7 DC7 or ground In AC mode7 there is a capacitor between the input connector and the vertical system circuit This keeps any constant DC level from entering the scope7 and all you see is the time varying ie AC part If you put the scope on DC7 then the constant voltage level also shows up If the input coupling is grounded7 then you force the input level to zero7 and this shows you where zero is on the screen Make sure that the scope is on auto trigger if you ground the input7 otherwise you will not see a trace Sometimes7 you also get to choose the input impedance for each channel Choosing the high input impedance usually 1 M9 is best if you want to measure voltage levels and not have the oscilloscope interact with the circuit However7 the oscilloscope will get a lot of use looking at fast pulsed signals transmitted down coaxial cable7 and you don7t want an impedance mismatch to cause the signal to be re ected back See Sec312 Cables with 509 characteristic impedances are very common in this work7 so you may nd a 509 input impedance option on the scope lf not7 you should use a tee connector on the input to put a 509 load in parallel with the input 54 OH 3 COMMON LABORATORY EQUIPMENT 353 Dual Trace Operation By ipping switches on the front you can look at either input channels trace separately or both at the same time There is obviously a problem though with viewing both simultaneously since the vertical trace can only be in one place at a time There are two ways to get around this One is the alternate the trace from channel one to channel two and back again This gives complete traces of each but doesn7t really show them to you at the same time lfthe signals are very repetitive and you7re not interested in ne detail this is okay However if you really want to see the traces at the same time select the chop option Here the trace jumps back and forth between the channels at some high frequency and you let your eye interpolate between the jumps If the sweep speed is relatively slow the interpolation is no problem and you probably cant tell the difference between alternate and chop However at high sweep speed the effect of the chopping action will be obvious 354 Bandwidth You should realize by now that high frequency operation gets hard and the oscilloscope gets more complicated and expensive Probably the single most important speci cation for an oscilloscope is its bandwidth and you will see that number printed on the front face right near the screen The number tells you the frequency at which a sine wave would appear only 71 as large as it should be You cannot trust the scope at frequencies approaching or exceeding the bandwidth Most of the scopes in the lab have 20 MHZ or 60 MHZ bandwidths A fast oscilloscope will have a bandwidth of a few hundred MHZ or more You will nd that you can the sweep speed over a large range but never much more than Bandwidth 1 The vertical sensitivity can be set independently of the sweep speed but scopes in general cannot go below around 2 mVdivision 36 DIGITIZERS 55 355 XY Operation On most oscilloscopes if you turn the sweep speed down to the lowest value one more notch puts the scope in the XY display mode Now the trace displays channel one X on the horizontal axis and channel two Y on the vertical For periodic signals the trace is a lissajous pattern from which you can determine the relative phase of the two inputs Oscilloscopes are also used this way as displays for various pieces of equip ment which have XY output options Thus the oscilloscope can be used as a plotting device in some cases 36 Digitizers Computers have become common in everyday life and the experimental physics laboratory is no exception In order to measure a voltage and deal with the result in a computer the voltage must be digitized The generic de vice that does this is the Analog to Digital Converter or ADC ADC7s come in approximately an in nite number of varieties and connect to computers in lots of different ways We will cover the particulars when we discuss the individual experiments but for now we will review some of the basics 361 ADC S Probably the most important speci cation for an ADC is its resolution We specify the resolution in terms of the number of binary digits bits that the ADC spreads out over its measuring range The actual measuring range can be varied externally by some circuit so the number of bits tells you how nely you can chop that range up Obviously the larger the number of bits the closer you can get to knowing exactly what the input voltage was before it was digitized A low resolution ADC will have 8 bits or less That is it divides the input voltage up into 256 pieces and gives the computer a number between 0 and 255 which represents the voltage A high resolution ADC has 16 bits or more 56 OH 3 COMMON LABORATORY EQUIPMENT High resolution does not come for free In the rst place it can mean a lot more data to handle For example if you want to histogram the voltage being measured with an 8 bit ADC then you need 256 channels for each histogram However if you want to make full use of a 16 bit ADC every histogram would have to consume 65536 channels That can use up computer memory and disk space in a hurry Resolution also affects the speed at which a voltage can be digitized Generally speaking it takes much less time to digitize a voltage into a smaller number of bits than it does for a large number of bits There are three general classes of ADC7s which I refer to as Flash Peak Voltage Sensing and Charge Integrating ADC7s A Flash ADC or waveform recorder simply reads the voltage level at its input and converts that voltage level into a number They are typically low resolution but run very fast Today you can get an 8 bit Flash ADC which digitizes at 100 Mhz ie one measurement every 10 ns This is fast enough so that just about any time varying signal can be converted to numbers so that a true representation of the signal can be stored in a computer To get better resolution you need to decide what it is about the signal you are really interested in For example if you only care about the maximum voltage value you can use a peak sensing ADC which digitizes the maximum voltage observed during some speci ed time Sometimes you are interested instead in the area underneath some voltage signal This is the case for example in elementary particle detectors where the net charge delivered is a measure of the particles energy For applications like this you can use an integrating ADC which digitzes the net charge absorbed over some time period ie 12 Vtdt where R is the resistance at the input For either of these types you can buy commercial ADC7s that digitize into 12 or 13 bits in 5 as or longer but remember that faster and more bits costs more money Don7t forget that one of your jobs as an experimenter will be to calibrate or otherwise know how to convert the number you get from an ADC into an actual voltage or charge value You will need to do this for some of the experiments in this course 36 DIGITIZERS 57 362 Other Digital Devices The opposite of an ADC is a DAC or Digital to Analog Converter Here the computer feeds the DAC a number depending on the number of bits and the DAC puts out an analog voltage proportional to that number The simplest DAC has just one bit and its output is either on or off In this case we refer to the device as an output register These devices are a way to control external equipment in an essentially computer independent fashion In many cases you want to digitize a time interval instead of a voltage level In the old days this was a two step process involving a device called a Time to Analog Converter TAC followed by an ADC Nowadays both these functions are packaged in a single device called a TDC The rules and ranges are very similar as for ADC7s Devices known as latches or input registers will take an external logic level and digitize the result into a single bit These are useful for telling whether some device is on or off or perhaps if something has happened which the computer should know about For the latter the computer interface circuit has to be able to interrupt what the computer is doing to let it know that something important happened on the outside 363 Dead Time Why should you care how fast an ADC or some other device digitizes Obviously the faster the device works the faster you can take data In fact this can be the limiting factor for many kinds of high sensitivity experiments When a device is busy digitizing it cannot deal with more input We refer to the cumulative time a device is busy as dead time Suppose 739 is the time needed to digitize an input pulse and R0 is the presumably random rate at which pulses are delivered to the digitizer lf Rm is the measured rate then in a time T the number of digitized pulses is MT The dead time incurred in time T is therefore RmTT so the number of pulses lost is RmTTR0 The total number of pulses delivered BOT must equal the number digitized 58 OH 3 COMMON LABORATORY EQUIPMENT plus the number lost7 so ROT RmT RmTTRQ and therefore R0 Rm 7 32 1 7R0 Rm 7 33 or R0 1 7 TRm The normal way to operate a digitizer is so that it can keep up with the rate at which pulses come in In other words7 the rate at which it digitizes 17 should be much greater than the rate at which pulses are delivered7 that is 7R0 ltlt 1 Equation 32 shows that in this case7 Rm R0 that is7 the measured rate is very close to the true rate7 which is just what you want Futhermore7 an accurate correction to the measured rate is given by Eq 33 which can be written as R0 Rm1 TRm under normal operation On the other hand7 if TRO gtgt 17 then Rm 17 That is7 the digitizier measures a pulse and before it can catch its breath7 another pulse comes along The device is always dead 7 and the measured rate is just one per digitizing time unit Essentially all information on the true rate is lost7 because the denominator of Eq 33 is close to zero You would have to know the value of 739 very precisely in order to make a correction that gives you the true rate 37 Digital Oscilloscopes The digital oscilloscope is a wonderful device Instead of taking the input voltage and feeding it directly onto the de ection plates of a CRT Fig 327 a digital oscilloscope rst digitizes the input signal using a Flash ADC7 stores the waveform in some internal memory7 and then has other circuitry to read that memory and display the output on the CRT At rst glance7 that may sound silly7 since we get the same result but in a much more roundabout way They key7 however7 is that we have the voltage stored as numbers7 and the 3 7 DIGITAL OSCILLOSCOPES 59 internal computer in the digital oscilloscope can do just about anything with the numbers Even though it works very differently from analog oscilloscopes7 digital scopes have controls that make them look as much like analog scopes as possible The same terminology is used7 and just about any function that is found on an analog scope will also be found on a digital one Digital oscilloscopes are relatively new7 and in this case TektroniX does not have a corner on the market In our lab7 for example7 we use the oscil loscopes by LeCroy and by Hewlett Packard For the models we have7 the LeCroy scopes are the most powerful 371 The LeCroy 9310 Digital Oscilloscope Our laboratory is equipped with LeCroy model 9310 and 9310A oscilloscopes The bandwidth of the 9310 is 300 Mhz and digitizes into 10k channels7 while the 9310A is 400 Mhz into 50k Both have a maximum digitizing rate of 100 Msamplessec These scopes differ in other minor ways7 but both are equipped with a complete mathematics library including Fast Fourier Transform and a PC compatible 35 inch oppy disk drive for data storage We chose these oscilloscopes partly because of how straightforward it is to use them Given experience with analog oscilloscopes7 you will have no trouble using these much more sophisticated devices You can do a lot by using very few of the features Most of the controls are menu driven7 and allow you to do any one of a number of things with the data Very simply7 you can stop the scope at any time and consider the last trace it threw up on the CRT Using the cursors7 you can read on the screen the values of Vt to within the resolution of the ADC 8 bits You can also read the time scale7 so you can do a better job estimating signal periods and frequencies The real power of the oscilloscope is realized with the internal math soft ware7 allowing you to do much more complicated things with the data You can take functions of the trace7 such as log Vt to see if Vt is consistent 60 OH 3 COMMON LABORATORY EQUIPMENT with an exponential decay You can even take the Fourier transform of the voltage as it comes in7 and measure the amplitude and phase of the different Fourier components For anything but the simplest data taking7 you should use the oppy drive to store traces for further analysis Start with an empty7 or just formatted7 15 Mb HD 35 inch oppy7 and follow these steps Insert the disk into the drive mounted on top of the scope o Press the Utilities button to bring up that menu Select Floppy disk utilities under that menu You will be asked to Reread the disk7 again from the menu 0 For the rst time you use the disk on that particular scope7 Press Perform disk format You will be asked to con rm that by pressing it again 7 Press Copy template to disk This puts a template le on the disk that identi es the properties of that particular oscilloscope At this point7 the disk has a directory le on it which containes the template le All storage operations go to that directory 0 Press Return three times to clear all the menus To store a particular trace7 it is a good idea to make sure the oscilloscope is Stopped 7 ie7 no longer updating traces Then bring up the Waveform store menu7 and choose the trace 17 27 or A D you want to store7 as well as the medium you want to store it to Disk A le with a name like sc1000 will be written to the disk7 where the name stands for store channel 1 if you in fact chose to store the trace corresponding to channel 1 and the extension keeps track of the number of times you stored that channel These traces are binary les that must be decoded elsewhere7 most easily on the PC in the laboratory The les produced by the oscilloscope are in binary format to save space Remember7 the default saves 10k real numbers 50k for the 9310A plus 38 COMPUTER INTERFACES 61 additional information for each trace To convert these binary les to ascii information you need the program 94TRAN which is supplied by LeCroy The use of binary les in general is described in a readme document also supplied by LeCroy These les are kept in the LeCroy subdirectory on the general use PC in our lab As described in readme the basic way to translate the le to a list of ascii values representing the voltage value for each point of the trace is through the command 94TRAN t l6tpl o lel239s le abc where letpl is the template le lelz39s is the output le and leabc is the binary le created by the oscilloscope For more detail you can also type 94TRAN h77 for help To get different information about that trace use a particular format77 speci cation le See readme For example if you want to get all the parameter settings of the scope when that trace was saved use the le afmt 94TRAN t l6tpl o lel239s fafmt leabc This is a good way to check the way the scope was setup but it is a good idea to write down the important things in your logbook when you take your data as some of the different parameter names are pretty cryptic The readme le has details on how to write your own format les if you want Once the data is in ascii form you can use anything you want to analyze it For example you might use MATLAB as described in Sec 143 and elsewhere in these notes 38 Computer Interfaces We7ve talked about digitizing devices like ADC7s on a very elementary scale and also more sophisticated digital instruments like oscilloscopes and multi meters In the end you want to get the data collected by these devices into a 62 OH 3 COMMON LABORATORY EQUIPMENT computer What7s more7 you want the computer to be able to control these devices The connection between the computer and the external device is done through an interface There are a huge number of different kinds of interfaces The architecture of an interface falls into one of two categories A serial interface is the simplest Here the computer communicates one bit at a time with the outside world The external device responds to a particular pattern of ones and zero7s7 and so does the computer Data is transferred between the two one bit at a time as well The connection is almost always done through a standard RS 232 serial line7 the same way a keyboard is attached to the computer Lots of pieces of this scheme are standard7 such as the communications software and even the connectors7 and this is a big advantage The problem7 of course7 is communication rate A fast serial line runs at 19200 bits per second the baud rate 7 and at this speed it would take over two minutes to read all 10K7 8 bit data points in one trace of the LeCroy 9310 If you are willing to give up the nice7 standard features of an RS 232 connection7 you can go faster but the interface hardware and software is more complicated In order to go faster7 the serial architecture is abandoned altogether7 and one goes to a parallel type of interface In this scheme7 many bits are transferred at the same time over a parallel set of wires The wires are connected through some kind of plug in card directly to the backplane77 of the computer7 and this really speeds things up The software for a particular computer can be rather simple as well Unfortunately7 you lose the ability to have some kind of standard interface because there are lots of different kinds of computers out there7 so both hardware and software can be very different Even the IBMPC and its look alikes have at least two distinct backplane architectures One way people have tried to bridge the gap between fast but speci c parallel interfaces and standard but slow serial interfaces is to build parallel middleman77 interfaces7 and hope that they become popular enough to be an industry standard That is7 a device like an ADC or meter might be designed to connect to the middleman7 and computer interfaces would also be designed to connect to it as well This potentially gives you more freedom of choice7 assuming that people out there provide you with lots of choices on 38 COMPUTER INTERFACES 63 both the device side and on the computer side Of course the middleman costs money by itself so this solution is generally more expensive Some examples of this type of interface are the following GPIB or General Purpose Interface Bus Also known as the lEEE 488 standard or as HPlB by people at Hewlett Packard corporation this has become quite popular in recent years It uses an ASCII code to communicate very similar to most serial line communication systems but uses a 24 pin connector allowing data to be transferred in parallel at some level It can transmit up to 1 MByte per second within this communication protocol CAMAC or Computer Automated Measurement And Control This standard has been around for a long long time and many people are hooked on it because they7ve already purchased lots of devices that connect to it It uses a rigid protocol called the Dataway for communication and data transfers can be quite fast and exible Programming in CAMAC is rather dif cult however and people usually end up buying commercial CAMAC software for their favorite computer FASTBUS This architecture was developed originally as a modern re placement for CAMAC particularly for very high rate and high density ap plications It is being used heavily at several large modern laboratories However it is rather costly and its popularity has been somewhat limited VME or Versa Module Europa Developed by a consortium of com mercial companies VME maps device locations directly into computer mem ory and is designed for high speed computer intensive applications Data transfer is very ef cient and the speed is around 20 MBytes per second It is becoming increasingly popular particularly in Europe 64 OH 3 COMMON LABORATORY EQUIPMENT 39 Exercises 1 An electromagnet is designed so that a 5 V potential difference drives 100 A through the coils The magnet is an effective inductor with an inductance L of 10 mH Your laboratory is short on space7 so you put the DC power supply across the room with the power cables along the wall You notice that the meter on the power supply has to be set to 6 V in order to get 5 V at the magnet On the other hand7 you are nowhere near the limit of the supply7 so it is happy to give you the power you need ls there any reason for you to be concerned Where did that volt go7 and what are the implications If there is something to be concerned about7 suggest a solution 2 You are given a low voltage7 high current power supply to use for an experiment The manual switch on the power supply is broken The power supply is kind of old7 and it looks like someone accidently hit the switch with a hammer and broke it off You replace the switch with something you found around the lab7 and it works the rst time7 but never again When you take it apart7 the contacts seem to be welded together7 and you know it wasnt that way when you put it in What happened Hint Recall that the voltage drop across an inductor is Ldidt and assume the switch disconnects the circuit over 1 msec or so 3 The following table is from the Tektronix Corp 1994 catalog selection guide for some of their oscilloscopes Model Bandwidth Sample Rate Resolution Time Bases 2232 100 MHZ 100 MSs 8 bits Dual 2221A 100 MHZ 100 MSs 8 bits Single 2212 60 MHZ 20 MSs 8 bits Single 2201 20 MHZ 10 MSs 8 bits Single You are looking at the output of a waveform generator on one of these oscilloscopes The generator is set to give a i2 V sine wave output If the sine wave period is set at 1 usec7 the scope indeed shows a 2 V amplitude 3 9 EXERCISES 65 However if the the period is 20 nsec the amplitude is 1 V Assuming the oscilloscope is not broken which one are you using 4 You want to measure the energies of various photons emitted in a nuclear decay The energies vary from 80 keV to 25 MeV but you want to measure two particular lines that are separated by 1 keV If you do this by digitizing the output of your energy detector at least how many bits does your ADC need to have 5 Pulses emitted randomly by a detector are studied on an oscilloscope The vertical sensitivity is 100 mVdiv and the sweep rate is 20 nsdiv The bandwidth of the scope is 400 MHZ The start of the sweep precedes the trigger point by 10 ns and the input impedence is 509 9 F7 0 El Estimate the pulse risetime What could you say about the risetime if the bandwidth were 40 MHz Estimate the trigger level These pulses are fed into a charge integrating ADC also with 509 input impedence The integration gate into the ADC is 100 ns long and precedes the pulses by 10 ns Sketch the spectrum shape digitized by the ADC Label the horizontal axis assuming i pC of integrated charge corresponds to one channel The ADC can digitize be read out by the computer and reset in 100 ns Estimate the number of counts in the spectrum after 100 sec if the average pulse rate is 1 kHz What is the number of counts if the rate is 1 MHz 66 OH 3 COMMON LABORATORY EQUIPMENT 6 A detector system measures the photon emission rate of a weak light source The photons are emitted randomly The system measures a rate of 10 kHz7 but the associated electronics requires 10 psec to register a photon7 and the system will not respond during that time What is the true rate at which the detector observes photons Ch4 Experiment 1 The Voltage Divider Now7s a good time to make some measurements based on what you7ve learned so far We will do some simple things with the voltage divider circuit7 in cluding both resistors and capacitors Circuits are most easily put together on a breadboard This is a at7 multilayered surface with holes in which you stick the leads of wires7 resistors7 capacitors7 and so on The holes are connected internally across on the component pads7 and downward on the power pads You can play around with a DMM and measure the resistance between different holes to convince yourself of the connections Dont forget to write everything down in your log book 41 The Resistor String Use a DMM to measure the voltage across the terminals of one of the small DC power supplies Switch the DMM to measure the current out of the terminals Do you suspect the supply is voltage or current regulated 67 68 CH 4 EXPERIMENT 1 THE VOLTAGE DIVIDER Connect two 1 K9 resistors in series on the breadboard and then connect the terminals of the power supply to each end of this two resistor string Once again measure the current across the output of the terminals Also measure the current through the string You will have to change the way you connect the leads of the DMM Now connect two more 1 K9 resistors in series with the others Move the connections from the power supply so that once again it is connected to each end of the string Repeat your voltage and current measurements Explain what you have seen so far Compare the results to Ohm7s law ls the power supply voltage or current regulated How well Can you estimate the equivalent internal resistance of the power supply Now measure the voltage drop across each of the four resistors Compare the result to what you expect based on the voltage divider relation Use your data and Ohm7s law to measure the resistance of each of the resistors Do you need to remeasure the current through each resistor Compare the resistance values you measure with the nominal value Remove the DC power supply and replace it with a waveform genera tor Set the waveform to a sine wave Use an oscilloscope to compare the voltage as a function of time across the resistor string from the waveform generator with the voltage across one of the resistors Put each of these into the two channels of the oscilloscope and trigger the scope on the channel corresponding to the waveform generator output Look at both traces simul taneously on either chop or alternate and compare the relative amplitudes of the input77 sine wave across the string and the output77 sine wave across the single resistor Discuss what you7ve measured You may want to try any number of variations on this theme For example put some of the resistors in parallel or series and see what you get Remember that Ohm7s law should always be valid and you can verify that anywhere you want in your circuit Also remember that the power supply supplies power If you hook up some other resistor to the circuit use what you7ve learned to calculate the power P Z39ZR VZR dissipated in that resistor and make sure it does not exceed the resistors power rating a m ADDING A CAPACITOR 69 Figure 4 1 Measuring gain and relative phase on an oscilloscope 42 Adding a Capacitor Now connect a resistor and capacitor in series Choose a resistance R and capacitance C so that the inverse time constant 1220 is well within the frequency range of the waveform generator and the oscilloscope Just as you did for the the resistor string measure the amplitude of the voltage across either the resistor or capacitor relative to waveform generator spanning well on either side of 1220 Also measure the phase of the output sine wave relative to the input sine wave Figure 4 1 shows how to make these measurements on the oscilloscope CRT using the circuit shown Refer to Fig 2 s for interpreting the input and output waveforms in terms of gain and phase it would be a good idea to set your frequency values logarithmically in 70 CH 4 EXPERIMENT 1 THE VOLTAGE DIVIDER stead of linearly That is instead of setting frequencies like VLo VLO AV VLO 2AV VH1 use something like VLO7 f gtlt VLO7 f2 gtlt Wop VH1 Make a clear table of your measurements and plot the gain ie the relative amplitudes and the relative phase as a function of frequency Think about how you want to scale the axes Making both axes linear is the worst choice Don7t forget that you measure frequency V but most of the relations we7ve derived are in terms of the angular frequency w 27w Compare your results to the calculated gain and phase difference Adding this to the plot would be a good idea Do you expect the same thing whether you were measuring the voltage across the capacitor or the resistor You can test this by changing the position of the oscilloscope probes in Fig 41 A sample of data and calculation is plotted in Fig 42 This plot was produced using MATLAB using the following commands load vdcapdat omegavdcap1 gain vdcap 2 phasevdcap3 R1453E3 CO1E 6 omegaflogspace1575 gainf 1sqrt1omegafRC 2 phasef180piatanomegafRC subplot211 loglogomegagain o omegafgainf aXiS1E2 1E7 2E4 2 Xlabel Angular Frequency Hz ylabel Gain subplot212 semilogxomegaphase o omegafphasef 42 ADDING A CAPACITOR 71 0 10 39 E 8102 102 103 10 105 106 107 Angular Frequency Hz o 80 o o g 60 E n 40 20 102 103 10 105 106 107 Angular Frequency Hz Figure 42 Sample of data on gain and phase shift with an RC voltage divider 72 CH 4 EXPERIMENT 1 THE VOLTAGE DIVIDER axis1E2 1E7 O 95 Xlabel Angular Frequency Hz ylabel Phase print dps vdcapps clear all The angular frequency7 gain7 and phase were all calculated separately and stored in the ascii le vdcapdat in three columns The curves were calculated using the known values of the resistor 1453 k9 and capacitor 01 ME Some more advanced plotting commands were used here7 to make log log and semilog plots7 and to put two plots on a single page 43 Response to a Pulse Use the waveform generator as a pulse generator and study the output using your RC voltage divider circuit Compare the input and output pulse shapes as a function of the width At of the pulse What happens if At gtgt RC What about At ltlt RC7 Ch5 Experiment 2 The Ramsauer Effect This is a simple and elegant experiment in quantum mechanical scattering You will show that when electrons at one particular energy impinge on xenon atoms7 they pass right through as if the atom was not there The experiment is described in detail in the following references 0 Demonstration of the Ramsduer Townsend E eet in a Xenon Thytratron7 Stephen G Kukolich7 American Journal of Physics 361968701 0 An Emtensz39on of the Ramsduer Townsend E eet in a Xenon Thyratron7 G A Woolsey7 American Journal of Physics 391971558 For more information on the physics associated with quantum mechanical matter wave transmission7 see 0 Introduction to the Structure of Matter7 John J Brehm and William J Mullin7 John Wiley and Sons 19897 Chapter Five 73 74 CH 5 EXPERIMENT 2 THE RAMSA UER EFFECT 0 Introductory Quantum Mechanics Richard L Liboff Second Edition Addison Wesley 1992 Section 78 0 Quantum Physics Robert Eisberg and Robert Resnick John Wiley and Sons Second Edition 1985 Chapter Six 0 Does the Spherical Step Potential Well Ecchibit the Ramsauer Townsend E ect R C Greenhow American Journal of Physics 61199323 The concepts of mean free path and cross section and how they pertain to the motion of particles in a gas are described very well in 0 Physics Robert Resnick David Halliday and Kenneth Krane John Wiley and Sons Fourth Edition 1992 Chapters 23 and 24 You may also want to consult Appendix B 51 Scattering from a Potential Well The Ramsauer effect sometimes called the Ramsauer Townsend Effect demon strates the difference between classical mechanics and quantum mechanics in the simple problem of a particle scattering from a potential energy well We mainly consider the problem in one dimension but make a few comments about the three dimensional case 511 Transmission past a One Dimensional Well Figure 51 summarizes the situation1 A particle is incident from the left 1The textbooks by BrehmampMullin Liboff EisbergampResnick and others all treat this or similar cases at appropriate levels of detail Other cases include the potential barrier as opposed to the well and the step function where the height of the potential energy changes abruptly 51 SCATTERING FROM A POTENTIAL WELL 75 Aeiklx Ceikgz Feiklz Beiiklx De ik E a V 0 f I II III V V0 Figure 51 A particle incident on a potential energy well where the potential energy is zero lts total mechanical energy ie kinetic plus potential energy is E7 which is constant with time Let7s rst consider what happens classically Conservation of energy de termines the motion through the equation Vz E 51 The function V is zero everywhere except for 7a S x S a where it is equal to 7V5 The particle is incident from the left and has a momentum p x2mE lt maintains this momentum until it gets to the well at z 7a where its momentum abruptly changes to p 2mE Vb Next it continues to the right hand edge of the well where the momentum changes back to p x 2mE Finally7 the particle continues on its way to the right forever The basic idea of quantum physics7 however7 is that particles can behave as waves with a wavelength A hp7 where Planck7s constant h 6626 gtlt 10 34 J sec414 gtlt 10 15 eV sec The motion of the particle is governed by the wave function ww with the quantity zbxzbxdz interpreted as the probability of nding the particle between z and z dz The wave function 76 CH 5 EXPERIMENT 2 THE RAMSA UER EFFECT is determined by solving Schrodinger7s wave equation 32mm 2m dz VWWW B74496 52 where h E h27T Equation 52 is easy to solve The quantity E 7 Vz is positive every where7 so we can write it as 6121006 2 Cm 7 7k W 53 hzkz where EVb for iagzga 2m 2 2 and hk E elsewhere 2m You7ve seen this equation lots of times before The rst time was probably when you studied the harmonic oscillator7 and learned that the solution is either sinkz or coskx with the appropriate integration constants We will use complex numbers see Appendix 04 to write the solution instead as 714 A6451 Be lkm 54 Let7s stop here for a moment and think about this Remember that 714 is supposed to represent the wave that is the particle The wavelength A of this wave7 by Eqn 54 is just 27rk Therefore the requirements on k listed in Eq 53 just state that hzkz h 2 1 2 i 4ltgt E7Vm 2m A 2m 2m which is just conservation of energy all over again The Schrodinger equation is just a statement of conservation of energy for a wavy particle Now lets go back to the wave function 714 in Eq 54 and see what it implies about the particles motion The time dependence of the wave is given by 6 so the term proportional to 6 represents a wave moving to the right and 6 is a wave moving to the left Divide the ziaxis into 51 SCATTERING FROM A POTENTIAL WELL 77 three regions7 namely regions I x S 7a7 II 7a 3 x S a7 and III x 2 a See Eqn 51 We have k k1 in regions I and III7 and k kg in region II7 where k1 and k2 are de ned in Eq 53 We write 7z for each of the three regions as Region I aw Ae k1m Be lklm Region II wuw Ce k Deilm Region III wu z Fe k1m We do not include a leftward moving wave in region III since we assume there are no more changes in potential past the well so the particle cannot turn around and come back There is already a key dz erenee between the classical and quantum me chanical treatments The solution allows for some portion of the incident wave to be re ected7 from the well That is7 B need not be zero7 and in fact generally is not This is clearly different from the classical case where the particle would always travel on past the well7 albeit with greater momentum for the time it is in the well Now the wave function and its rst derivative must be continuous every where This allows us to determine relations between A7 B7 0 D7 and F by matching awe and 71x at z ia These four conditions give us A671k1aB6lk1a 0671k2aD61k2a zklAe lkla 7 zleerl ZkZCe ZkZ 7 ZkZDerZ Oelkga D671k2a F6lk1a szCerZ 7 szDe ZkZ zleerl These are four equations in ve unknowns A fth relation would just deter mine the normalization of the wave function7 but we wont bother with this here Let7s calculate the probability that an incident particle makes it past the well The amplitude of the incident wave is A and the amplitude of the transmitted wave is F Therefore7 the transmission probability T is given by T wma WAP AVI A A 78 CH 5 EXPERIMENT 2 THE RAMSA UER EFFECT It is pretty easy to solve for FA using the above relations Solve the rst two for A in terms of C and D by eliminating B Then solve the last two to get 0 and D in terms of F The result is A k2 k2 7 eZlkla cos 2km 7 3 1 2 2 Isls2 F sin 2h2a which leads to 1 1 kLkZ 2 i 1 71 2 sin22k2a T 4 m2 where k1 2mEh and k2 4 2mE lbh We can therefore write l i 1 lV 02s11122k a 5 5 T 4 EE 10 2 39 The re ection coef cient R lBlzlAlZ can also be calculated in the same way Can you think of a simpler way to do this7 having already calculated 7 The transmission coef cient T is plotted as a function of Elb in Fig 527 for a a 10h12mlb The transmission probability is unity only at certain values of the incident kinetic energy E This is wholly different from the classical case where transrnission would always occur Consider the physical interpretation of the points where T reaches unity This is when sin2 2h2a 0 or kga nir2 where n is any integer However7 k2 27rA2 where A2 is the wavelength of the particle while it is in the well Therefore7 the condition for T 1 is n2 2a That is7 there is perfect transmission past the well only when an integral number of half wavelengths ts perfectly inside the well Note that the width of the well is 2a 512 Three Dimensional Scattering Of course7 the experiment we will do involves scattering in three dimensions and we have only worked things out for the one dimensional case The generalization to three dimensions is the spherical well for which Vr 7V6 for r 3 17 but zero elsewhere The analysis of this case is somewhat 51 SCATTERING FROM A POTENTIAL WELL 79 Transmission Probability g Figure 52 Transmission probability for a square well The barrier width is chosen so that kgaz 1001 Elb more complicated7 and we wont treat it here Nevertheless7 the essential point still remains7 namely that the well becomes invisible to the incident particle when ka n7r2 where thZQm E Vb When we talk about scattering in three dimensions7 the language becomes a bit specialized ln particular7 we talk about the scattering cross section77 which measures the probability that an incident particle scatters from some target In this experiment7 you observe the total cross section as opposed to a dz erentz39al cross section which measures the probability that the particle scatters into any direction at all For classical scattering of a point particle from a hard sphere77 of radius 17 the total cross section is just given by the cross sectional area of the sphere7 namely 7Ta2 When the well becomes transparent to the incident particle7 the total cross section vanishes Analysis of the three dimensional case shows that when ka n7r27 the cross section passes through a resonance That is7 the phase of the scattered wave7 relative to the incident wave7 passes through 90 80 CH 5 EXPERIMENT 2 THE RAMSA UER EFFECT There is an important difference between the one dimensional and three dimensional cases This is that only the rst resonance ie when the con dition ha 7r2 is met is clearly visible Therefore you epeet to see only one dip in the cross section in this measurement The paper by Greenhow provides some interesting although somewhat advanced reading You should review the material on three dimensional scattering in books like Brehm and Mullin or Liboff before getting into it in detail Greenhow actually analyzed the case of the perfect spherical well and shows that the Ramsauer effect is generated but only in a restricted way The real potential of the xenon atom of course is considerably more complicated than a spherical well but this nevertheless serves as a convenient and worthwhile approximation 52 Measurements Your measurements are very similar to those originally performed by Ram sauer that is you will be scattering electrons from xenon gas atoms The procedure we use is based closely on the experiment described by Kukolich The idea is shown schematically in Fig 53 using a gure borrowed from Kukolich Electrons are released by a hot lament and made to accelerate to some energy E by a voltage V so that E eV where e 1602 gtlt10 19 C Electrons which scatter from the xenon atoms in their path move off in some direction and likely hit the shield a conductor which transports the elec trons back to ground potential On the other hand the electrons which make it through without scattering eventually strike the plate which also con ducts the electrons back to ground You will determine the behavior of the scattering cross section by measuring the plate current relative to the shield current as a function of V A large small scattering cross section therefore corresponds to a small large plate current The actual setup is diagrammed in Fig 54 The acceleration and scatter ing take place in a xenon lled electron tube called a 2D21 thyratron You make connections to the various internal components through pins num bered in Fig 54 on the tube For your convenience the tube plugs into a socket wired to a labeled panel with banana plug connectors Electrons 52 MEASUREMENTS 81 Shield Xe atoms Cathode Grid Plate Figure 53 Schematic diagram of Ramsauer Effect apparatus Electrons are accelerated towards the plate7 where they are collected if they do not scatter from Xenon atoms Otherwise7 they are collected by the shield or grid Vi l V 10km 2D21 6 Vs W 1009 Figure 54 Setup used to measure the Ramsauer Effect 82 CH 5 EXPERIMENT 2 THE RAMSA UER EFFECT that are captured by the shield or the plate are returned to ground through the resistors on the respective circuit7 and you determine the shield or plate currents from the voltage drop across these resistors These resistors are in a breadboard7 and you should consider different values for them and test that the currents you deduce are the same Since the plate current is typically much less than shield current7 you generally want the plate resistor to be much larger than the shield resistor so that their voltage drops are compara ble Suggested starting values are 10 k9 and 100 Q for the plate and shield resistors respectively 521 Procedure The data taking procedure is straightforward First7 you need to heat the cathode lament in the thyratron so that it emits electrons This is done using a standard laboratory DC voltage supply and a high current voltage divider to send a speci c current through the lament The lament is con nected to pins 3 and 4 of the thyratron7 and you get the right current with a voltage of about 4 V Adjust the voltage divider and power supply so that you get 4 V before connecting to the pins Too much voltage can damage the lament and the tube becomes useless You should monitor this voltage throughout the data taking procedure to make sure it does not change Measure the voltages at the plate and at the shield as a function of the applied voltage V Adjust V through the voltage divider connected to another DC voltage supply You should vary V in relatively small steps between 0 and around 5 V You should nd that the plate current 2391 passes through a maximum of 015 MA or so for V N 1 V This is the Ramsauer Effect The plate current is a maximum because the scattering cross section has gotten very small allowing a large number of electrons to pass through the xenon gas and strike the plate Sample data7 taken from Kukolich7 is shown by the open points and solid line in Fig 55 In order to get quantitative results7 some more work needs to be done First7 you must realize that the thyratron is a pretty weird electron acceler ator As you change the value of V7 the electric eld lines inside change and the probability that electrons get to the plate will certainly change7 regard 5 2 MEASUREMENTS in i INN xinou my cuanw u um um Iu u am a z 1 e a mum IcLELERIKmG one man Figuze 5 5 Sample of low data taken fzom Kuko ich less of whethei oi not theie is gas inside ln feet how do you know fol Sule that the maximum in t e plete ouuent couesponds at all E39iXE seetteiing7 Symbolioslly the plate and shield ouuents ole iel t d MV zeVfV1 PeaMEN EL Wheze Pee4T is a function of the election enezgy and shou d become the Ramsauez E ect iesonsnoe and fV is geometzical teotoi depending on the eooeleieting Voltage and the details of the th lotion The pxoblem is that you do not know befozehand how to sepeiete the effects of fV and P50ATE Howeyei you can easily sepaxate these effects using you eppeietus After tummy 0 the lament uoltaya and lattmg the lament eool down dunk the top of the tube in liquid nitiogen is fleezes out the xenon and zeduces the bulb pxessuxe to a negligible level Repeat the meesuiehnents above a d sinoe Pee4T 0 you deteihnine fV h m wheze the 2 indicate measuze merits taken with the xenon xemoved Figs of V uie 5 5 also plots 2 as a function 84 CH 5 EXPERIMENT 2 THE RAMSA UER EFFECT 522 Analysis When analyzing your data realize that the electrons are accelerated by the potential difference between the the negative terminal of the power supply V and the the shield and that VS changes with V Their energy is therefore given by E e V 7 VS corrections where there are still some additional corrections These are studied in more detail in Sec 53 These corrections amount to about 04 V which should be added in before calculating E Plot PSCAT as a function of the incident electron momentum p 2mE where m is the electron mass To compare with the gures in Kukolich realize that they ignore any extraneous factors and compute momentum simply as xV 7 V5 also ignoring any other corrections Different experiments show that the radius of the xenon atom is around 4 Calculate the well depth of the xenon atom potential assuming that it is approximated by a spherical well with this radius Does this sound reasonable to you Because the electrons scatter the electron beam intensity diminishes ex ponentially as a function of the distance traveled that is IQ Ioe wLSCAT where LSCAT is the mean free path77 through the gas in the tube lnsomuch as the plate current measures the beam intensity at the plate the scattering probability PSCAT is related to the mean free path by e LLSCAT 1 7 PSCAT where L is the distance through the tube to the plate For the 2D2l thyra tron L 07 cm This information can be used to estimate the scattering cross section 039 since it is related to the mean free path by LSCAT lpna where pn is the number of xenon atoms per unit volume Determine pn from the ideal gas law2 using the quoted pressure of 005 Torr for the 2D2l at room 2The ideal gas law says that 731 NkT where N is the total number of atoms in the volume 1 hence p7 Nvi See for example Resnick Halliday and Kranei 53 ADVANCED TOPICS 85 temperature Compare the calculated cross section on and off resonance with the geometric cross section 7Ta2 where a is the radius of the xenon atom To summarize7 you can calculate the following quantities from your data 0 The approximate well depth Vb of the xenon atom o The scattering probability7 which can be compared to the literature 0 The scattering cross section7 on and off resonance 53 Advanced Topics As discussed by Kukolich7 there is a discrepancy between the observed value of V where the minimum cross section occurs7 and that found in the liter ature He attributes this to a 04 V contact potential7 but Woolsey shows that this is in fact both from the contact potential and from the thermal energy of the electrons when they emerge from the lament You can show this in the same way as Woolsey You will use the same apparatus as for the standard77 measurements above7 but with some simple rearrangements The lament of the 2D21 is made of barium oxide and the shield is made of nickel Since the nickel has the higher work function of the two7 there is a contact potential difference that causes electrons to spontaneously ow from the lament to the shield7 even if V 7 V5 is zero Therefore7 the actual energy of the electrons is somewhat higher than you would expect from V7 VS alone Call that contact potential difference V0 There is another reason that the electrons are higher energy than you would rst expect The lament is hot7 so the electrons have some thermal energy when they are emitted As dictated by statistical mechanics7 this thermal energy is not one single value but instead is distributed over a range of energies The appropriate distribution function is the Maxwell Boltzmann distribution which says that the number of electrons with energy ETH is proportional to e ETHkT where T is the temperature of the lament The average energy of the electrons is ETH 3kT2 See for example7 Resnick7 Halliday7 and Krane 86 CH 5 EXPERIMENT 2 THE RAMSA UER EFFECT So the incident energy of the electrons is given by E6VV9VCV 56 where 6V ETH represents the average effect of the thermal electron distri bution Now the issue is how do we measure V0 and V The key is to realize that when the xenon in the thyratron tube is frozen out the plate current will behave like see Woolsey z z oe BVRETZV 57 where VRET is a retarding voltage between the shield and the cathode That is as VRET increases it makes it harder for electrons to get to the shield The fact that z is a nite value equal to 2390 when there is no potential difference between the shield and lament VRET 0 just indicates that electrons still ow to the shield due to their thermal energy As the retarding voltage is increased the shield current goes down exponentially This continues until the retarding voltage equals the contact voltage after which the current decreases even more rapidly due to space charge saturation at the cathode See Woolsey for more details The procedure is therefore straightforward With the top of the thyratron dunked in liquid nitrogen as before reverse the polarity of V by switching around the connections As you increase V from zero record the shield voltage V5 You may need to nd a more precise voltmeter than the standard DMM7s used in the lab If you plot 2 VsRS versus V VS on semilog paper then the slope of the line gives you V according to Eq 57 At some value of V the data will abruptly change and 2 will fall more rapidly At this value of V you determine V0 V V5 This is shown in Fig 56 which is taken from Woolsey7s paper Take several measurements of this type Try changing the cathode l ament voltage by a volt or so around the standard value of 4 V This will change the temperature of the lament so it should change the slope ac cordingly The contact potential on the other hand should be unaffected Use measurements of this type to determine V0 and V and to estimate their uncertainties Use your results and Eq 56 to reanalyze the Ramsauer effect How does this affect your determination of the well depth What about the cross section determination 5 3 ADVANCED TOPICS 4n ru a a mu mm o mm Figure 56 Sample of data with reversed polarity taken from Woolsey 88 CH 5 EXPERIMENT 2 THE RAMSA UER EFFECT Ch6 Experimental Uncertainties Before we go on to do more experiments7 we need to learn one of the most important things there is about making measurements Every measurement yields some number Of equal and sometimes greater importance is the uncertainty with which we know how close that number approximates the right answer In this chapter7 we will learn the basic facts about estimating and reporting experimental uncertainties Sometimes people refer to experimental error when they mean experi mental uncertainty This is unfortunate7 since error implies that a mistake was made somewhere7 and that is not what we are talking about here This terminology is pretty well ingrained into the jargon of experiments7 though7 so you might as well get used to it When an experimenter quotes the result of a measurement7 the uncer tainty in that result should also be quoted As you will see7 the measurement result will give a sort of central value of some quantity7 call it Q7 and the uncertainty gives some idea of how far on either side of Q you have to go to hit that true value We write the uncertainty in Q as 6Q and quote the result of the measurement as Qi Q You should always get used to writing down your results this way 89 90 CH 6 EXPERIMENTAL UN CERTAIN TIES We will discuss some of the basics of uncertainties and statistical anal ysis in this course In particular the concepts you need to carry out the experiments will be outlined and they are covered rather well in 0 Practical Physics G L Squires Third Edition Cambridge University Press 1991 However it is a good idea to have a more thorough reference on this stuff There are a lot of books out there but I recommend 0 An Introduction to Error Analysis The Study of Uncertainties in Phys ical Measurements John R Taylor University Science Books 1982 We will also discuss using MATLAB for some of the numberical manipulations commonly used for determining uncertainty Refer to Sec 143 for the basics on MATLAB including the main references 0 The Student Edition of MATLAB Prentice Hall 1994 0 Numerical Methods for Physics Alejandro Garcia Prentice Hall 1994 61 Systematic and Random Uncertainties There are two kinds of experimental uncertainty namely Systematic and Random Uncertainty Sometimes it can be hard to tell the difference because their meanings are not always precisely de ned 1 will give you some con venient ways to think about them but as with all things in Experimental Physics your intuition will get better with experience Systematic uncertainty comes from not knowing everything there is to know about your experiment If you could precisely duplicate the conditions every time you make a measurement then your systematic uncertainty would 61 SYSTEMATIC AND RANDOM UNCERTAINTIES 91 be zero However it is impossible to precisely duplicate things The room temperature will be different the positions of other people in the room or the building will have changed and the phase of the moon is not the same to name just a few Another possibility is that your measuring instrument is only accurate to some level and this may be the most important systematic uncertainty All of these things can affect your measurement at some level and one of your jobs is to try and estimate how big the effect can be Some guidelines are in order for estimating systmatic uncertainty In very many cases one thing in particular may dominate the systematic un certainty Try to nd out what that thing is and estimate how much it may have changed your result That would be an estimate of your systematic un certainty You can go further perhaps and gure out how much it actually might have changed things It would cause the central value to shift and then you would apply a correction77 to your result How well can you make that correction Answer that question and you can get another estimate of your systematic uncertainty Of course if you want to make your experiment more and more precise the approach is to identify the sources of systematic uncertainty and reduce their effect somehow Random uncertainties are different At their most fundamental level they come from the chance uctuations of nature although in many cases the system is so complicated that you will observe uctuations that might as well be random The point is you cannot account for random uncertainty other than to calculate how big it is The key to random uncertainties is that if you make many measurements of the same quantity then the random uctuations will average to zero over many trials Obviously then the way to reduce random uncertainty is to make lots of measurements Because oftheir random nature this source of experimental uncertainty can be estimated quite precisely More on that soon Let7s try a simple example Suppose you want to measure the resistance of a 500 foot roll of 32 gage aluminum wire You just hook up your DMM to the ends of the wire on the spool and measure the resistance There is some uncertainty associated with how long the wire actually is so you measure many spools of wire to get an idea of how big the random uctuations are However there is also an uncertainty associated with the precision of the DMM No matter how many measurements you make that systematic 92 CH 6 EXPERIMENTAL UN CERTAIN TIES uncertainty will always be present Lets get more precise about these things 62 Determining the Uncertainty Remember that by their nature systematic and random uncertainties are treated differently In particular you can only estimate the systematic un certainty We7ll discuss some ways to do that but as with just about every thing in Experimental Physics practice makes perfect On the other hand you can deal with random uncertainties in well de ned ways and well go through those 621 Systematic Uncertainty Try looking for systematic uncertainties in two places First consider the accuracy of your measuring instruments This includes meters clocks rulers digitizers oscilloscopes and so on How precisely can you read the device in the rst place If a ruler is graduated in 1 mm increments for example you cant measure the length of something much better than that Does you clock tick off in seconds If so it is hard to argue that you could measure the time it takes something to happen any more precisely Also keep in mind the manufacturers speci cations How accurately does your oscilloscope measure voltage How well do they guarantee the conversion of charge to digits in a charge integrating ADC The second thing to keep in mind is the effect external factors have on your measurement For example suppose you are trying to precisely mea sure the length of something with a carefully graduated metal ruler but the room temperature is uctuating in a i5 C range The length of the ruler is given by L L0 aT 7 To where or is the metan thermal expansion coef cient Therefore the actual length L of your sample will only be known to a precision of Oz i25 O due to this systematic uncertainty There are an in nite number of examples of this sort of thing 62 DETERMINING THE UN CERTAIN TY 93 Don t make the mistake of assuming you will gure all these things out when you are analyzing your eccperiment Record anything you suspect might be important Try to nd out what you can about your instruments as well 622 Random Uncertainty The idea of random uncertainty is that the uncertainty will average away with a large number of trials Consequently you would expect the average value of a number of measurements to closely approximate the true value at least within the limit of any systematic uncertainty This in fact is the case and we will talk more about it when we discuss statistical analysis in a later chapter However if the average approximates the true value how do we calculate the magnitude of the random uncertainty Let7s make some de nitions and then I will tell you what to interpret as the random uncertainty Suppose you make n measurements of a quantity d and the result is the list of numbers 1 2 x We de ne the mean i also written as ltdgt of the measurements to be 1 V L i 7 Mean Value 61 n i1 That is i is just the average value of z from the measurements The variance 02 of the measurements is de ned to be o 7 if Variance 62 and obviously has something to do with how far the values uctuate about the mean value Don7t worry about the n 7 1 in the dominator instead of just n We7ll discuss this later as well The quantity om the square root of the variance is called the standard deviation You can show that the variance can also be written as 2 7 922 63 039 m n71 This form is particularly useful for programming computers since you can calculate both x2 and i within the same loop 94 CH 6 EXPERIMENTAL UN CERTAIN TIES Now as we discussed above7 we interpret the mean i as our best approx imation to the true value77 of x Furthermore7 we interpret the standard deviation 0x as the uncertainty in each measurement xi On the other hand7 as we will show in Sec 6317 the uncertainty in the mean value of the xi as it approximates the true value of x is given by 03 Tm 64 So7 when you report the result of a series of measurements of a you write i i 02 That is7 the random uncertainty in the measured value is 02 Don7t forget that these formulas apply only to random uncertainties7 and do not apply to systematic uncertainties You can always minimize the ran dom uncertainty by taking lots of measurements and averaging them to gether However7 if systematic uncertainties dominate7 then the total uncer tainty in the measurement will be bigger than that given by 64 623 Using MATLAB MATLAB can be very useful for your data analysis needs Given a list of numbers read into a vector array x see Sec 1437 you can easily determine7 for example7 an array xsq corresponding to the squares of these elements xsqx 2 The 77 before the exponentiation symbol indicates that the operation is to be performed element by element7 as opposed to calculating the square of a matrix This notation is used for all element by element operations The program also has simple functions available which directly calculate many of the quantities needed here For example7 nlengthx 63 PROPAGATION OF ERRORS 95 xsumsumx Xbarmeanx sigxstdx return the number of elements in the array x the sum of the values the mean of the values and the standard deviation of the values Various other functions return the maximum max value minimum min value median median value and the product ofthe elements prod ln MATLAB language for example the standard deviation can also be calculated from the sequence of commands nlengthx xbarmeanx xsigsqrt sum X Xbar 2 nD This should return precisely the same value you would get using the std function Of course this is just the tip of the iceberg We will point out the most relevant functions as we go along but don7t forget there are lots more that we won7t mention Consult the MATLAB Users Guide for more information 63 Propagation of Errors If you measure some value x with an uncertainty 6x but you are interested in some quantity q which is a function of x ie q qz then what is the corresponding uncertainty 6g For example suppose the gain 9 of an ampli er depends on voltage V as g AV If the voltage is known to within 6V how well do we know 9 Suppose things are more complicated and q is a function of two inde pendently measured quantities z and y q qxy An example might be determining the temperature T from a gas bulb thermometer with volume 1 and pressure P through the ideal gas law T PvNR How do you deter 96 CH 6 EXPERIMENTAL UN CERTAIN TIES 10 D4 0 f Figure 61 Propagation of errors for a single independent variable mine the uncertainty in T from the uncertainty in P and 1 or7 in general7 6g frorn 6x and 6y All this is accomplished through propagation of errors This phrase is so ingrained in the scienti c cornrnunity7 that l won7t bother substituting uncertainty for error In any case7 the prescription is straightforward Let7s consider the single variable case rst Figure 61 schernatically plots the quantity q qx as a function of x Say the best value for z is 0 Then7 the best value for q is go qz0 As shown in the gure7 the uncertainty in x 6x is related to the uncertainty in q just by the slope of the curve at z x0 That is7 dq Ch gtlt 6x 65 Sq mo gives the uncertainty in q The absolute value insures that the result is a positive number Now let q be a function of several variables7 ie q ay The best value for q is go qx0y07 7 and there will be contributions to the 63 PROPAGATION OF ERRORS 97 uncertainty 6g from each variable following from Eqn 65 6Q 6 7 q M gtlt6z qu y X 6y 110 m0 The big question though is how to combine the Sq to get 6g Do we simply add them together ie 6g 6th 6qy 7 This might seem unfairly large since if z uctuates all the way to its maximum uncertainty so that z x0 6x then it is unlikely that y would uctuate that much as well and so on In fact you might think that if z and y are correlated then an upward uctuation in z might imply there is a good chance that y uctuates downward In this case you are temped to use something like 6g loqmioqyl In general there is no clear answer to this question It depends on the speci c nature of the uncertainties whether they are random or systematic and whether or not they are correlated with each other There is however one speci c case where there is a straightforward answer This is the case where all uncertainties are random and uncorrelated and the answer is 6a 62 my MF 3g 2 3g a f In this case we say that the uncertainties are added in quadrature 2 2 6y 66 110 Even though Eqn 66 only applies to random uncorrelated uncertainties it is often used incorrectly in other circumstances Probably the most dangerous incorrect use is for random uncertainties which are not completely uncorrelated You should at least convince yourself that the variables x y and so forth are independent to at least a good approximation There is a method which can take into account correlations of random uncertainties and we will discuss it in a later chapter Adding errors in quadrature is almost always incorrect for systematic uncertainties and you should do the best you can to estimate their net effect One practice is to quote the random and systematic errors separately ie q 10 i 6QlRANDOM i qlSYSTEMATIC 98 CH 6 EXPERIMENTAL UN CERTAIN TIES so you can at least let the reader know their relative contributions You should always keep in mind the relative sizes ofthe terms in Eqn 66 If any of the g3a 6 are signi cantly bigger than the rest then it will dominate the net uncertainty especially since you add the squares1 In this case you may be able to think of that variable as the only important one as far as the uncertainty is concerned Many experiments to measure some quantity more precisely than it has been done before are based on ideas that can reduce the dominant uncertainty 631 Examples Fractional Uncertainty We will work out some general formulas for propagating uncertainties In the cases for more than one variable we assume that errors add in quadrature Power Law of One Variable Consider the earlier example of gain as a function of voltage ie g AV where we know the voltage V to within i V Using Eqn 65 we have 69 nAVnil V Notice however that there is a simpler way to write this namely 69 6V 7 n7 67 g V lt gt That is the fractional uncertainty in g is just n times the fractional uncer tainty in V This is true for any power law relation q az where 04 and 6 are arbitrary constants that is 6 6x 109519 i 357 q z 1Donlt be swayed by the notation 621 It is just a simple and common shorthand for 61 63 PROPAGATION OF ERRORS 99 Sum of Two Variables Consider the general case q AzBy where A and B are arbitrary constants Equation 66 tells us that 6g A262x Bz zy 68 In this case7 there is no simple form for the fractional error in q General Power Law Product Now look at the general case q Army Again using Eqn 66 we have 6g mAmmilyn gt262z nAzmyn l gt2 5 but it is obviously simpler to write 6 6x 2 6 2 g 1 9 Knowing the fractional uncertainties in L y and so on makes it simple to see if any of them dominate the result 69 Two simple but useful cases of Eqn 69 are q zy and q In both cases7 the fractional uncertainty in q is the sum in quadrature of the fractional uncertainties in z and y The Uncertainty in the Mean Back in Sec 622 we just quoted the result for the uncertainty in the mean We can now derive it using propagation of errors Start with the de nition of the mean value Eq 61 100 CH 6 EXPERIMENTAL UN CERTAIN TIES Here n is a constant and we determine the uncertainty in the mean simply by applying Eq 68 12 12 12 6x 76x176x276n n n 71 Now the supposition in Sec 622 was that the m are all separate measure ments of the same quantity x and that the uncertainty in z is given by the standard deviation 039 Therefore7 all the terms in this equation are the same7 and we have which proves Eq 64 632 Dominant Uncertainty If two or more quantities are measured to determine the value of some de rived result7 their individual uncertainties all contribute to the uncertainty in the nal value If one of the uncertainty in one of those quantities makes the largest contribution to the nal uncertainty7 we refer to it as the dom inant uncertainty It is smart to identify the dominant source or sources of uncertainty in an experiment That7s the one you want to learn how to measure better Doing a better job on the others might be nice7 but it won7t buy you a signi cantly more precise result in the end The relative precision of each of the quantities is not all that matters You also need to know how that quantity contributes in the end Equation 69 makes this point particularly clear If one of the quantities enters with some large exponent7 then that exponent ampli es the contribution of its uncer tainty Even though ixs may be smaller than lgy z may dominate the uncertainty in the end if m is much larger than n 64 EXERCISES 101 64 Exercises 1 You measure the following voltages across some resistor with a three digit DMM As far as you know7 nothing is changing so all the measurements are supposed to be of the same quantity VR 231 235 226 222 230 227 229 233 225 229 Determine the best value of VR from the mean of the measurements 9 F7 What systematic uncertainty would you assign to the measurements 0 Assuming the uctuations are random7 determine the random uncer tainty from the standard deviation Somebody comes along and tells you that the true value of VB is 223 What can you conclude El 2 From Squires In the following examples7 q is a given function of the independent measured quantities z and y Calculate the value of q and its uncertainty Sq assuming the uncertainties are all independent and random7 from the given values and uncertainties for z and y a qx2for25i1 b qz72yfor100i3andy45i2 c qzlny forx1000i006 andy100i2 d q17for50i2 3 Police use radar guns to catch speeders The guns measure the frequency f of radio waves re ected off of cars moving with speed c This differs from the emitted frequency f0 because of the Doppler effect we 102 CH 6 EXPERIMENTAL UN CERTAIN TIES for a car moving away at speed 1 What fractional uncertainty must the radar guns achieve to measure a car7s speed to 1 mph 4 The period T of a pendulum is related to its length L by the relation T27TZ 9 where g is the acceleration due to gravity Suppose you are measuring 9 from the period and length of a particular pendulum You have measured the length of the pendulum to be 11325i00014 m You independently measure the period to within an uncertainty of 0067 that is STT 6 gtlt 104 What is the fractional uncertainty ie uncertainty in 97 assuming that the uncertainties in L and T are independent and random 5 You have a rod of some metal and you are changing its temperature T A sensitive gauge measures the deviation of the rod from its nominal length l 1500000 m Assuming the rod expands linearly with temperature7 you want to determine the coef cient of linear expansion 04 ie the change in length per degree K7 and the actual length l0 before any temperature change is applied The measurements of the length deviation Al as a function of the temperature change AT are as follows AT Al Mm AT Al Mm AT Al Mm 08 70 22 110 36 130 10 110 26 150 38 170 12 130 28 120 42 160 16 100 30 130 44 190 18 130 34 160 50 160 Plot the points and draw three straight lines through them 0 The line that best seems to go through the points 0 The line with the largest reasonable slope o The line with the smallest possible slope 64 EXERCISES 103 Use your own estimates by eye to determine these lines Don7t use a tting program Use the slopes and the intercepts ofthese lines to determine ai a and l0 1 6 Suppose you wish to measure the gravitational acceleration g by using something like the Galileo experiment That is7 you drop an object from some height h and you know that the distance it falls in a time t is given by gtz For a given experimental run7 the fractional uncertainty in h is ShI1 4 and the fractional uncertainty in t is Stt 15 Find the fractional uncertainty in g from this data7 assuming the uncertainties are random and uncorrelated 7 You want to measure the value of an inductor L First7 you measure the voltage V across a resistor B when 121 i 004 mA ows through it and nd V 253 i 008 V Then7 you measure the decay time 739 in an RC dircuit with this resistor and a capacitor C and get 739 RC 0463 i 0006 msec Finally7 you hook the capacitor up to the inductor and measure the oscillator frequency w Um 136 i 9 kHz What is the value of L and its uncertainty 8 A simple pendulum is used to measure the gravitational acceleration g The period T of the pendulum is given by T27T g 1lsin2 g 4 2 for a pendulum initially released from rest at an angle 00 Note that T a 27139 Lg as 00 a 0 The pendulum length is L 872 i 06 cm The period is determined by measuring the total time for 100 round trip swings A total time of 192 sec is measured7 but the clock cannot be read to better than i100 ms What is the period and its uncertainty Neglecting the effect of a nite value of 00 determine 9 and its uncer tainty from this data Assume uncorrelated7 random uncertainties 9 F7 You are told that the pendulum is released from an angle less than 10 What is the systematic uncertainty in g from this information 0 El Which entity the timing clock7 the length measurement7 or the un known release angle limits the precision of the measurement 104 CH 6 EXPERIMENTAL UN CERTAIN TIES 9 The decay asymmetry A of the neutron has been measured by Bopp etal PhysRevLett 561986919 who nd 2M1 7 A 701146 i 00019 1 3 This value is perfectly consistent with but more precise than earlier results The neutron lifetime 739 has also been measured by several groups and the results are not entirely consistent with each other The lifetime is given by 7 51637 560 T 1 3 and has been measured to be 918 i 14 560 by Christenson etal PhysRevD519721628 881 i 8 560 by Bondarenko etal JETP Lett 281978303 937 i 18 560 by Byrne etal PhysLett 92B1980274 and 8876 i 30 560 by Mampe etal PhysRevLett 631989593 Which if any of the measurements of 739 are consistent with the result for A Which if any of the measurements of 739 are inconsistent with the result for A Explain your answers A plot may help Ch7 Experiment 3 Gravitational Acceleration This is a conceptually simple experiment We will measure the value of 97 the acceleration due to gravity7 from the period of a pendulum The main point is to determine 9 and understand the uncertainty If you measure it precisely enough7 you can see the effect of the Earth7s shape You can also convince yourself that Einstein7s Principle of Equivalence77 is valid The physics and technique are straightforward7 and can be found in just about any introductory physics textbook Most of the interesting stuff is neatly collected in o Handbook of Physics7 E U Condon and Hugh Odishaw7 McGraw Hill Book Company7 Part ll7 Chapter 77 pg57 59 71 Gravity and the Pendulum According to lore7 Galileo rst pointed out that all objects fall at the same ac celeration7 independent of their mass This is pretty much true7 at least near the surface of the earth We understand this simply in terms of Newtonian 105 106 CH 7 EXPERIMENT 3 GRAVITATIONAL ACCELERATION mechanics which says that V F mi 71 and Newtonian gravity which says that mME F G 72 Bi where m is the mass of the object ME is the mass of the earth and RE is the radius of the earth which we assume is much larger than the height from which the object is dropped In other words the acceleration a due to gravity near the earth7s surface which we call 9 is g Gig 98 msec2 73 E In fact since the earth is atter near the poles and therefore closer to the center of the earth there is some variation with lattitude At sea level one nds g 9780524 msec2 at the equator and g 9832329 msec2 at the poles a fractional difference of about one half of one percent There are practical as well as philosophical reason to know the value of g with high precision For example oil exploration can exploit small changes in the gravitational acceleration due to underground density changes Consequently there has been a lot of work over the years aimed at high quality measurements of 9 Until very nifty techniques based on measuring the rate of free fall using interferometry came into beingl the pendulum was the best method We will explore that technique in this laboratory A sketch of the physical pendulum and its approximation as a simple pen dulum are shown in Fig 71 For a precise measurement of 9 it is important to realize that no pendulum is truly simple so we7ll start with the physical pendulum The rotational inertia I E frzdm is de ned around the pivot point and L is the distance from the pivot to the center of mass Newton7s Second Law in terms of the swing angle 6 is d26 7 7 dt2 where the torque is T lW gtlt Ll MgLsin6 1See Practical Physics Gr Li Squires Third Edition Cambridge 1985 71 GRAVITY AND THE PENDULUM 107 Figure 71 Physical and simple pendula The physical pendulum realizes the size and mass distribution with a rotational inertia I about the pivot point If approximated as a simple pendulurn7 ie a point rnass suspended on a rnassless string7 then I MLZ 108 CH 7 EXPERIMENT 3 GRAVITATIONAL ACCELERATION so the equation of motion can be written 126 MgL dt2 I This is generally solved in the small angle approximation that is7 by setting sint9 0 In this case7 we are reduced to simple harmonic motion with angular frequency sin 6 0 74 i w E 2 Physical Pendulum 75 The approximation as a simple pendulum just sets I MLZ7 so we have A E 2 Simple Pendulum 76 One goal of this experiment is measure the pendulum period precisely enough to see a departure from the small angle approximation This depar ture can be calculated theoretically We nd a rst integral of the motion77 by rst multiplying Eq 74 by dQdt d6d26 2d6 if i 00 dt dtZ 1 dt m then rearranging the derivatives to get d 2 a wzcos 0 2 1 d6 7 7 LUZ cos 6 constant which implies that 2dt The constant2 can be determined by assuming the pendulum is released from rest dQdt 0 at an angle 00 ie constant 7212 cos 1 Therefore 2 2w2cost9 7 cos 1 2This constant can in fact be expressed in terms of the total mechanical energy 71 GRAVITY AND THE PENDULUM 109 and so 90 d6 T l wxE lt7 77 0 cos 6 7 cos 1 2 4 where the period is T and we realize it takes one fourth of a period to move to the vertical position from the point of release This integral cannot be solved analytically7 but we can make use of some mathematical trickery and expand it in powers of 00 Since cos 1 7 2sin2x2 we can rewrite Eq 77 as 90 d0 LOT 0 sin2902 7 sum2 5 and then make a change of variables to sinx sin02sin002 which leads us to 7r2 dx wT l 7 0 1 7 sin2002 sin2 x 2 4 Now we can easily expand the integrand in powers of sin2002 l l 9 l 1 Esin230sin2x 17 sin20 sin2 4 2 and carry out the integral term by term The result is 1 i 2 t9 W1Zs1n2 0 78 Ti W where w is given by Eq 75 or Eq 76 The small angle approximation is clearly recovered as 60 7 0 The second term in Eq 787 which we might call the rst order correction 7 is small but you should be able to con rm it in this experiment 711 Principle of Equivalence Einstein realized that there was some cheating going on when we derived Eq 73 using Eq 71 and Eq 72 The mass M of the object in question 110 CH 7 EXPERIMENT 3 GRAVITATIONAL ACCELERATION is used in two very different ways and we just assumed they were the same thing without asking why In Eq 71 Newton7s Second Law mass is just the proportionality constant that connects acceleration a precisely de ned kinematic quantity with a new and more mysterious quantity called force In Eq 72 Newton7s Law of Gravity we use M to mean the quantity that gives rise to a gravitational force77 in the rst place We should actually write the two masses differently ie inertial mass77 M1 for Newton7s Second Law and gravitational mass77 MC for Newton7s Law of Gravity We should therefore reduce the physical pendulum to the simple pendu lum by writing I MILZ whereas the torque is more properly written as 739 MagL sin 0 The period for the simple pendulum in the small angle approximation becomes 1 L M 5 T 27139 771 9 MG You might then ask ls the gravitational mass the same as the inertial mass for all materials7 and test the answer by measuring the period for pendulum bobs made from different stuff Clearly if you are going to test whether Einstein was right or not you must be prepared to make as accurate a measurement as possible The best limit3 on lMI 7 MalMI was obtained by Eric Adelberger and collaborators at the University of Washington They obtained lMI 7 MalM1 lt 10 M using a torsion balance Early in this century however a limit of lt 3 gtlt 10 6 was obtained with a simple pendulum 72 Measurements and Analysis The technique is simple and straightforward but you have to take some care because the point is to make precise measurements Set up a pendulum by hanging a massive bob from a exible but inelastic 3See Gravitation and Spacetz39me Hans Ci Ohanian and Remo Ruf ni Second Edition Norton 1994 72 MEASUREMENTS AND ANALYSIS 111 line You want to keep your physical77 pendulum as simple77 as possible7 so make sure the line is very lightweight and the bob is small and massive You still will have to be careful when you determine the pendulum length The length of line determines the period7 so pick something convenient A couple of meters is a good place to start Timing the period precisely is very important Set up the pendulum so the bob swings close to the oor or table top Put a mark on the surface under the bob when it is motionless You7ll use this mark to time the period as the pendulum swings past it Set the pendulum in motion and use the digital stopwatch to time the period The stopwatch reads in 001 second intervals and the period will likely be a couple of seconds That is7 you would immediately have a systematic uncertainty of 005 by timing one swing That7s not good enough7 since we are trying to measure 9 to 01 or so7 which means we need to know the period at least twice as well7 or 005 However7 you can easily reduce the systematic uncertainty by a factor of 10 by timing 10 swings instead of only one Figure 72 histograms the period as determined4 in several runs of ten swings each This analysis is done in MATLAB simply by entering the mea surements into an array7 de ning another array to set the histogram bins7 using the command hist to sort the data7 and the command stairs to make the plot There is some scatter in the measurements which probably comes from human response time in starting and stopping the stopwatch We will treat this scatter as a random uncertainty7 that is we can take the period as the average of all these N measurements with an uncertainty given by the standard deviation divided by That is7 the period is determined to be T 27994 i 000097 a 003 measurement Determine the length of the pendulum as best you can Assign an uncer tainty to the length7 and calculate g from Eq 76 and T 27Tw Determine 69 the uncertainty in g by propagating the errors from the length L and period T ls your result for g i 69 clearly within the established polar and equatorial values 4Data taken by Jason Castro7 Shaker High School Class of 1996 112 CH 7 EXPERIMENT 3 GRAVITATIONAL ACCELERATION 8 MEAN2799 7 STD 0005386 Number of measurements 279 2795 28 2805 281 Period Figure 72 Histogram of several measurements ofthe pendulum period7 each made by timing ten swings and dividing by ten to reduce the systematic uncertainty from reading the stopwatch 72 MEASUREMENTS AND ANALYSIS 113 Try to con rm the rst order correction in Eq 78 by changing the angle 00 and plot T i 6T as a function of sin2602 You will have to use an angle 00 that causes a correction signi cantly larger than your measurement uncertainty You can measure 00 accurately enough just be putting a ruled scale on the oor or table top7 and use trigonometry to turn the point at which you release the pendulum into an angle 00 Do you determine a straight line with the correct slope 114 CH 7 EXPERIMENT 3 GRAVITATIONAL ACCELERATION Ch8 Experiment 4 Dielectric Constants of Gases This experiment measures the dielectric constant of some gases This is a simple physical property of materials and in this case it can be related to the way the electron charge is distributed in atoms or molecules that make up the gas The technique is simple and is an instructive way to measure quantities that differ from each other by only a small amount The basic physics involved is rather straightforward For a good basic discussion of the fundamentals you might review 0 Physics Robert Resnick David Halliday and Kenneth Krane John Wiley and Sons Fourth Edition 1992 Chap22 24 The Ideal Gas Law Chap31 Capacitors arid Dielectrics Chap38 Electromagnetic Oscillations A ne discussion of the electronic properties of gases and how they give rise to the dielectric constant can be found in 115 116 CH 8 EXPERIMENT 4 DIELECTRIC CONSTANTS OF GASES o The Feynman Lectures on Physics7 R Feynman7 R Leighton7 and M Sands7 Addison Wesley 19647 Volll Chapt11 The measurement will be made using the beat method77 of measuring fre quency This is discussed in 0 Practical Physics7 G L Squires7 Third Edition Cambridge University Press 19917 Sec66 You will also likely use some edition of the Handbook of Chemistry and Physics to look up dielectric constants7 ionization potentials7 and dipole mo ments for various gases Note also that an experiment rather similar to this one is described in Y Kraftmakher7 Am J Phys 6419961209 81 Electrostatics of Gases The physics associated with this experiment is pretty simple It has to do with how charge can be stored in a capacitor7 and how the material inside the capacitor changes the amount of charge that can be stored After some review7 we will get into speci cs for the case where the material inside the capacitor is a gas Let7s review the traditional de nition of the dielectric constant We7ll start with a capacitor7 pictured as a pair of parallel plates7 separated by some distance that is small compared to their size Assume rst that the space in between the plates is a vacuum If the capacitor is charged up to some voltage V by a battery and a charge iqo is stored on the two plates q0 on one and egg on the other7 then the capacitance is de ned to be 00 QOV Now suppose that the space between the plates is lled with some non conducting material It turns out that if the capacitor is charged to the 81 ELECTROSTATICS OF GASES 117 same voltage V then more charge iq can be stored on the plates In other words the capacitance increases to C qV The increase in the capacitance de nes the dielectric constant H through 0 gt 1 8 1 H 7 00 Obviously Is also measures the increased stored charge if the plates are kept at constant potential ie H qqo The dielectric constant H is a property ofthe material and does not depend on the capacitor geometery or the voltage This is not at all obvious from these simple de nitions but we won7t go into it in any more detail here So why does the charge on the capacitor plates increase when the material is inserted The reason is that although the atoms or molecules that make up the material are electrically neutral the positive and negative charges in them are somewhat independent When they are inside the electric eld of the capacitor the negative charges tend to point towards the positive capacitor plate and vice versa This cancels out some of the electric eld However if the plates are kept at constant voltage the total electric eld inside must remain unchanged Therefore there is a buildup of charge on the plates and the capacitance increases When the positive and negative charges line up77 in this way in an atom or molecule it obtains a dipole moment For point charges of it separated by a distance z the dipole moment p qz See Resnick Halliday and Krane If the charge is not concentrated at a point but has some distribution in space as for an atom or molecule then the dipole moment comes from integrating the charge distribution weighted by the position The dielectric constant H can be directly related to the atomic or molecu lar dipole moment p The electric eld inbetween the plates of the capacitor is E 060 where 039 qA is the charge per unit area on the plates There are always some free77 charges supplied by the voltage source but with the dielectric in place there are also some polarization charges from the effect of the dipole moments The key is to realize that the polarization charge per 118 CH 8 EXPERIMENT 4 DIELECTRIC CONSTANTS OF GASES unit area is just given by the net dipole moment per unit volume called P See the Feynman Lectures Therefore the electric eld inside the capacitor is given by E g UFREE UPOL UFREE P 60 60 60 so that P E 1 7 UFREE 60 lt EOEgt Equation 81 then implies that 1 P 8 2 K 7 60E The task then is to relate the individual atomic or molecular dipole moments to the net dipole moment per unit volume How we do this depends on where those dipole moments come from and there are two ways that can happen Some molecules have permanent dipole moments They make up the class called polar dielectrics This happens because the atoms that make up the molecules are arranged in some asymmetric pattern and the atomic nuclei cause the charge to be redistributed in some way The most common example is the water molecule H20 where the atoms form a triangular shape with the oxygen at the vertex A permanent dipole moment forms along the line passing through the oxygen nucleus and which bisects the two hydrogen nuclei It is hard to calculate the magnitude of the dipole moment but you can look it up in the Handbook of Chemistry and Physics and you nd pH20 185 Debye 617 gtlt 10 30 Cm This is more or less typical of most polar molecules with values ranging from about a factor of ten smaller to a factor of ten larger Atoms and most molecules however have their electric charge symmetri cally distributed and do not have permanent electric dipole moments They can nevertheless have dielectric properties because the electric eld between the capacitor plates induces an electric dipole moment in them These ma terials are called nonpolar dielectrics and their behavior is considerably dif ferent from polar dielectrics The action with a nonpolar dielectric inside the capacitor plates is shown schematically in Fig 81 taken directly from Resnick Halliday and Krane which shows the effect on the electric eld 81 ELECTROSTATICS OF GASES 119 a b c E 0 Figure 81 a A slab of nonpolar dielectric material The circles represent neutral atoms of molecules b An external electric eld E0 displaces the positive and electric charges in the atom and induces a dipole moment These displaced charges induce charges of the opposite sign on the capacitor plates c7 increasing the stored charge in order to keep E0 unchanged Let7s rst estimate the dielectric constant of some gas made of nonpolar atoms We will use a very simple model7 namely where the electron is bound to the atom by some imaginary spring with spring constant k When the electron is placed in an electric eld E7 there is an electric force on it of magnitude 6E This causes the electron to be displaced a distance x where it is counterbalanced by the spring force kz It will make more sense to express the spring constant k in terms of the electron mass m and the angular frequency of the simple harmonic oscillations we namely k mwg Therefore 771ng 6E7 and the atomic dipole moment is i 62E mwg p ex 83 Before we go further7 it is instructive to estimate the size of this dipole moment Estimate we by assuming that blue hVO is the energy needed to ionize the atom This is a real seat of the pants estimate It takes something like 10 eV to ionize an atom7 so take we 10 eVh 152 gtlt 1016sec Let7s also take a relatively high electric eld7 say 100 V across a capacitor with a 1 mm gap7 or E 105 Vm Then we nd p 122 gtlt 120 CH 8 EXPERIMENT 4 DIELECTRIC CONSTANTS OF GASES 10 35 Cm You certainly expect therefore that the dielectric constant for a nonpolar gas should be a lot smaller than for a gas made of polar molecules There are however other important differences as we shall soon see Anyway let7s continue and estimate the dielectric constant for the non polar gas The dipole moment per unit volume is just P Np where N is the number of atoms or molecules per unit volume and p is given by Eq 83 Equation 82 then gives 2 H 1 E 1 N762 Nonpolar Gas 84 60E eomwo We approximate N from the ideal gas law namely N PkT 24 gtlt 1025m3 at room temperature T 300 K and atmospheric pressure 73 101 gtlt 105 Nmz Using the same seat of the pants estimate we nd that H is very close to unity in fact H 71 33 gtlt 104 This is surprisingly close to what is actually measured especially for such a very simple estimate Keep in mind however how H 7 1 depends on N and tag Lastly we will brie y derive the dielectric constant for a polar gas At rst we suspect that it should be a lot larger because the dipole moment is so much bigger but it isn7t quite as simple as that The permanent dipoles do indeed tend to line up along the electric eld but they are thermally agitated and don7t stay aligned very long because they are always bumping into each other See the Feynman Lectures or a book on statistical mechanics if you want to go through the derivation but for now I will just quote the result NZE E P p Ngtltpgtltltip gt 3kT 3kT This makes good sense qualitatively The effective dipole moment ofthe polar molecule ie PN is just the permanent dipole moment p reduced by the factor pEBkT which measures the electrostatic energy of dipole alignment ie pE roughly with respect to the thermal energy of the molecules kT roughly This reduction factor is signi cant For example water vapor p 617 gtlt 10 30 Cm at room temperature in a 105 Vm electric eld has a reduction factor of 5 gtlt 10 5 bringing it more in line with nonpolar dielectrics Putting this together into an expression for the dielectric constant gives N102 360kT Polar Gas 85 8 2 MEASUREMENTS 121 Figure 8 2 Sketch of an old fashioned Variable parallel plate capacitor Note that as for nonpolar gases s e 1 is proportional to N and therefore proportional to the pressure However for polar gases it is a strong function ftemperature On the other hand the dielectric constant for polar gases shows no dependence on the ionization potential i e m of the molecules 8 2 Measurements You should realize something right away For gases typical values for the dielectric constant a are very close to unity in fact a e 1 will be on the order of 1074 or so if you were to measure is directly therefore you would need a fractional experimental uncertainty SKK e 10o5 0 001 in order to get a 10 measurement o s e 1 This would be hard he trick is to come up with a way to measure se 1 directly We will do this by rst relating s to the frequency of electromagnetic oscillations and then by learning how to measure the dz emnca of two such frequencies The heart of the experiment is a variable parallel plate capacitor the kind that had been used to tune the frequency in old fashioned radios sketch of such a thing is shown in Fig 8 2 The relative surface area of the capacitance depends of course on what is between the plates because of the 122 CH 8 EXPERIMENT 4 DIELECTRIC CONSTANTS OF GASES dielectric constant of the material The capacitance C of the capacitor can be changed either by tuning it which makes a big change or by changing the gas between the plates a small change This capacitor is put in series with an inductor with inductance L forming an LC Oscillator See Halliday Resnick and Krane The cur rent in this circuit as well as the voltage across either the capacitor or the oscillator varies sinusoidally like cos wt where w Changing the capacitance then changes the angular frequency w Still however it is very hard to measure the dielectric constant by introducing a gas inbetween the plates and rerneasuring the frequency since the change in frequency would be very small Instead of measuring the frequency directly we will measure how much it changes using the method of beats Suppose you have two signals call them yl and y2 both with the same amplitude A but with different angular frequencies M and Luz If those two signals are added you nd 11 12 A cos w1t cos wgt 2A cos cos If an Luz then the addition signal oscillates with a angular frequency AD M Luz but with an amplitude that itself oscillates with a very low frequency lwl 7 wgl 2 These slow oscillations in the amplitude are called beats and it is not hard to build a circuit that gives an output signal which oscillates with the beat frequency of two input signals Okay so the beat frequency measures the difference between two numbers M and Luz which are very close to each other That is essentially what you want namely to measure the difference between the angular frequency with the capacitor in vacuum and the angular frequency with the capacitor in gas 1 HLC The problem though is that if the capacitor is in vacuum you don7t have the signal with it lled with gas and vice versal How can you get the two signals you want at the same time The solution to this problem is to have an external reference frequency and measure the change in the LC oscillator frequency relative to the refer ence In the experiment setup the external oscillator is packaged in a box 82 MEASUREMENTS 123 Internal OSCillatOr 7 Oscilloscope and Mixer Gas Inlet and Pumping port Figure 83 Setup for measuring the dielectric constant of a gas together with the circuit which forms the difference signal You connect the LC oscillator signal into the box7 and the output is a signal whose angu lar frequency is the di erence as opposed to half the difference of the two angular frequencies of the input 821 Procedure The setup is shown in Fig 83 The LG oscillator is inside another box which sits inside a bell jar With the bell jar removed7 you can tune the capacitance by adjusting the knob on the side of the box It is important to tune the capacitor so that the LC oscillator gives very closely the same frequency as the reference signal in the eccternal boa Do this by carefully turning the knob and watching the output difference signal on an oscilloscope The object is to make the difference signal frequency as small as possible after the bell jar is replaced and then evacuated This will likely take some trial and error Also7 be careful that the frequency of the LC oscillator is always less than the reference That is7 keep your eye on the difference frequency when you pump out the bell jar the frequency will change7 but you dont want it to go through zero You might notice that the difference frequency changes drastically while you7re trying to measure it In fact7 you7ll nd that as you bring some object7 124 CH 8 EXPERIMENT 4 DIELECTRIC CONSTANTS OF GASES like your hand near the bell jar you can change the difference frequency at will The reason is that you can disturb the electric eld and hence the capacitance of the variable capacitor without actually touching it You cant change the capacitance by very much but you7re going through all this so you can detect small changes in capacitance For this reason it is a good idea to cover the bell jar with a grounded conducting shell that shields the capacitor from external sources of noise This is pretty easily accomplished using a large sheet of aluminum foil connected with a wire to a ground point Once you7ve tuned the capacitor you need to measure the reference fre quency V0 too27139 In fact you cannot get at the internal oscillator that provides it but you can easily measure the frequency of the LC oscillator at this point just by hooking up its output leads to the oscilloscope If you tuned the capacitor perfectly then this would be exactly equal to the reference frequency Now evacuate the bell jar Using the valves connected to the lling hose let in air or one of the gases inside the pressurized gas bottles Let it in a little at a time and measure the difference frequency for each pressure The pressure gauge that we are using rneasures pressure 73 in inches of rner cury where one atmosphere is 30 inches Record the pressure and difference frequency at each setting 822 Analysis Don7t forget that the angular frequency w is 27w where V is the frequency you measure using the oscilloscope Let7s make some de nitions 0 we is the angular frequency of the external oscillator that is the refer ence frequency that you rneasured earlier 0 CO is the capacitance of the variable capacitor when the space inbetween the plates is evacuated o The dielectric constant H 1X where X is called the electric suscep tibilz39ty 82 MEASUREMENTS 125 Realize the X is a small number in this experiment Also realize that it is a function of the pressure 73 In fact7 according to Eqs 84 and 857 X is proportional to 73 We can write that W0 We 1 xLOe where 6we represents the small difference between the reference frequency and the evacuated frequency of the LC oscillator You should be able to have tuned the capacitor so that 6wewe 01 or smaller Measuring the difference frequency allows you to determine we 7w where w is the angular frequency of the LC oscillator7 whether or not there is some gas between the plates Therefore 1 weiw 77w6we xLOe lt1 6w0 xLOe 1 m we 1 77g 6we where we write 1 LCe we We can get away with this because it rnulti plies another very srnall nurnber7 namely 1 1 1 7 7 m 7 2X This introduces an uncertainty in X on the order of Swewe7 and it is unlikely that this small uncertainty will dominate the measurement If you plot your measurements as w 7 we versus 73 or PPATM then you should get a straight line In fact7 the slope of the line should give you weXATM2 if plotted against PPATM The yiintercept of the line tells you how closely you tuned the capacitor to the reference frequency Draw the best straight line that you can through your data points7 and call this the best value for the slope Also draw lines with the largest and smallest slopes you think are reasonable Use these lines to estimate the uncertainty in your measurement An example is shown in Fig 84 How does this uncertainty 126 CH 8 EXPERIMENT 4 DIELECTRIC CONSTANTS OF GASES quot an SLOPE PROPORTIONAL TO ELECTRIC SUSCEPTIBILITY AT 1 ATM O quot as quot m DIFFERENCE IN ANGULAR FREQUENCY kHz to N lb 04 06 08 PRESSURE N ATMOSPHERES Figure 84 Sample Data for the Dielectric Constant compare with that from the imprecise tuning of the variable capacitor How does your value compare with book77 values for the dielectric constant Is it within your experimental uncertainty Do this with a few different gases7 including the air Check to see if there is a correlation between the ionization potential of the gas and the dielectric constant You might check the humidity in the atmosphere on the day you do the measurement What effect does moisture in the air have on your result Remember that unlike N2 and 02 water is a polar molecule 83 Advanced Topics Get a bottle of Helium gas to try The electrons in a He atom are very tightly bound it takes 245 eV of energy to remove an electron Use this7 and the discussion of nonpolar dielectric constants7 to estimate the dielectric constant of Helium Compare this to your measurements If you are unable to determine a value for the He dielectric constant7 see if you can determine 83 ADVANCED TOPICS 127 an upper limit77 for it That is7 if you see no slope in your plot of we 7 w versus 73 how big a slope would you be able to put through the data ls this upper limit consistent with the book value Check the literature for possible polar gas molecules that you could mea sure Be careful7 since a lot of such molecules make explosive gases it would probably be wise to pick something less dangerous Try to vary the temper ature by cooling or heating the outside of the bell jar Do the same with a nonpolar gas like N2 or 002 Can you at least approximately verify the temperature dependence in Eqs 84 and 85 128 CH 8 EXPERIMENT 4 DIELECTRIC CONSTANTS OF GASES Ch9 Statistical Analysis We continue our discussion of uncertainties In this chapter we will be talking about random uncertainties only although some of the techniques we will develop like curve tting can be applied in more general cases As before refer to the books by Squires or Taylor for more details 0 Practical Physics G L Squires Third Edition Cambridge University Press 1991 0 An Introduction to Error Analysis The Study of Uncertainties in Phys ical Measurements John R Taylor University Science Books 1982 This chapter also begins a more serious discussion about datat analysis especially using computers As before we will make use of MATLAB for all examples and again I refer you to the following documentaion o The Student Edition of MATLAB Prentice Hall 1994 0 Numerical Methods for Physics Alejandro Garcia Prentice Hall 1994 See Sec 143 for more details 129 130 CH 9 STATISTICAL ANALYSIS 91 The Mean as the Best Value Let7s introduce the subject by reconsidering something we took for granted If we measure some quantity z a whole bunch of times then we assumed that the best approximation to the true value of x was the mean of m called f See Eq 61 This is in fact true and we can prove it Let A be the value that best approximates the true value of x Assume that we have 71 measurements of x called z and that each measurement has a standard deviation uncertainty 039 Consider the quantity X2A de ned as The conjecture is that A is the value that minimizes X2 This actually makes some sense since if A gets too far away from all the values then X2 gets very big We will put this conjecture on rmer ground at the end of this chapter when we talk about the Gaussian Distribution but for now lets take it at face value So lets minimize X2A with respect to A That is dXZ 2 n i IFA10 dA 02 l which implies Ex 7 A 0 11 i1 or n i1 And there it is The best approximation to the true value of x ie A is given by the mean of x From the de nition ofthe standard deviation it is clear that the minimum value of X2 is XTIIN E X2i 7quot 1 91 THE MEAN AS THE BEST VALUE 131 When we generalize the de nition of X2 the mean value will be very useful when evaluating data We7ll come back to this a few more times in this chapter This allows us to make a very useful generalized de nition of the mean7 called the weighted average lfthe measurements of z do not all have the same standard deviation uncertainty7 it doesn7t make sense to just take a straight average of all of them lnstead7 the values with the smallest uncertainties should be worth more7 somehow In this case7 X2AZ n 11 U1 where 01 is the standard deviation uncertainty of xi We can determine A in exactly the same way7 namely by setting dXZdA 0 We nd that 21 w i Az n 211 wi Weighted Average 91 where the weights wi E 1012 Obviously7 if all the weights are equal7 then Eq 91 reduces to Eq 61 The uncertainty in the weighted average can be derived using propagation of errors7 just as we did in the unweighted case You nd w 92 391 There are no built in functions like mean in MATLAB for the weighted average7 but it is pretty simple to either do it from the command line7 or write an appropriate m le to carry this out It could be done7 for example7 almost entirely within the sum command See the discussion in Sec 623 What about the minimum value of X2 for a weighted average That depends on the weights7 ie the individual uncertainties assigned to the measurements xi However7 if the various data points i 01 are indeed consistent with a single true value7 then you expect that x HN n 7 1 132 CH 9 STATISTICAL ANALYSIS 92 Curve Fitting You will very very often want to test your data against some model Plot ting your data in a suitable way can help you do that as we7ve discussed Sometimes however you need to be more precise In particular if the model depends on some parameters you want to vary those parameters so that the model ts your data This would give you the best value for those parameters Of course you also want to know with what uncertainty your data determines those parameters This is the subject of curve tting We can develop it just by following our prescription for showing that the best value for some quantity is given by the mean of the measured values To be sure we are developing the technique known as the method of least squares since the object is to minimize the sum of the squares of the deviations between the data and the tting function There are in fact other techniques such as the principle of maximum likelihood and multiple regression that may actually be better suited to some class of problems but we wont be discussing them here 921 Straight Line Fitting Lets start with the simplest generalization of our prescription for the mean Whereas the mean is a one parameter t to a set of values xi we now consider a two parameter t to a set of points In particular the model is a straight line of the form yaoa and our job is to nd the best values of a0 and a1 and their uncertainties as determined by the data If we were to x a1 0 then we should get 10 7 For now we assume all the values y have the same uncertainty we call Ty and we ignore any uncertainties in the z The X2 function is de ned just as before namely 2 i n i 00 i 012 X 00611 2 E 39 391 71 2 2 11 U21 l U 92 CURVE FITTING 133 which we minimze just as before namely 3X2 2 n 7 77 yi7a 7azi0 dag Ugi1 0 1 3X2 2 n 7 77 yi7a 7azizi0 601 Ugi1 0 1 which leads to a pair of equations for 10 and a1 V L V L 0071 11 i1 i1 a0 a12z 93 i1 i1 i1 From now on we will drop the limits 239 1 and n from the summation signs because it gets too crowded The solutions for 10 and 11 are simple 2 96 2 1M E 96139 Emu 10 A 71 E i E E A 2 A E n 7 So if you draw a line y a0a1x over a plot of your x y data the line will pass near all the points assuming that a straight line was a good approxi mation in the rst place You would likely derive some physical quantities from the values of a0 and a1 11 94 where Remember when we de ned the standard deviation Eq 62 Instead of dividing by n we divided by n 7 1 We won7t try to prove it but the reason is that i is not the true value of s but rather just our best estimate for x Therefore the uncertainty is actually a slight bit larger than it would have been if we used the true value and this shows up by dividing by n 71 instead of 71 If 71 gets to be very large then i is very close to the true value and n 7 1 is very close to n so this is at least consistent Now when we t to a straight line we have the same problem That is our data determine the best values for 10 and 11 not the true values In this 134 CH 9 STATISTICAL ANALYSIS case7 however7 there are two free parameters7 not one as for the simple mean The standard deviations Q are therefore given by 1 0 7172yi 00 al if and the minimum value of X2 is n 7 2 The number of data points minus the number of free parameters 2 for the straight line t and 1 for the simple mean is called the number of degrees of freedom Of course7 we need to know the uncertainties in 10 and 11 as well Equa tions 94 give 10 and al in terms of things we know the uncertainties for7 namely the yi Therefore7 just use propagation of errors to get what we want The result is no 2 7 and 0amp177 2 UZEzZ y l U i A no and so the result of your t should be reported as 10 i ago and 11 i Jul If the individual points do not all have the same uncertainty7 but instead are xi7y i 0 then the generalization is straightforward We have 96200701 Z wi i do i 012 where wi 101 The rest follows in the same way as above7 and the equations are listed in both Squires and Taylor They are also written out in Garcia7 including programs in MATLAB and in FORTRAN See the next section Using MATLAB to t straight lines Straight line tting is so common a problem that MATLAB has a built in function for this The function polyfitxy17 where x and y are arrays of the same length7 returns a two dimensional array which contains the slope and intercept of the best t straight line The third argument7 17 is a simple ex tension of this function I will explain it shortly Furthermore7 the function polyvapx returns the best t functional values7 ie the approximation to the y7 for the slope and intercept in p as returned by polyfit The following series of commands 92 CURVE FITTING 135 ppolyfitxy1 fpolyvalpx plotxy o xf ts the points to a straight line7 and then plots the data points themselves along with the t For unequally weighted points7 you cant really use polyfit However7 it is a simple matter to program such a thing using MATLAB7 and in fact Garcia has already provided us with an m le which does this It is called Iinregm7 and it is available via anonymous FTP from The Mathworksl7 or from me I also reproduce inregm in Fig 91 922 Fitting to Linear Functions If you wanted to t data to a parabola7 ie y a0 alz agzz you could follow the same procedure as for a straight line You would get three equations in the three unknowns 107 11 and a2 and solve them as before However7 there is something more profound going on There is an entire class of functions that can be t this way You might suspect as much if write Eq 93 as 71 E m 2 2 2 E iyi Shades of Math III This is called a system of linear equations 7 and there are very general ways of solving these things 00 01 Any function of the form 9 bif1 bzsz 39 39 39 amme can be t using the procedure as for a straight line The straight line7 of course7 is the case where m 27 f1z 17 and f2z x The parabola7 1You can get lots of free software like this from The Mathworksi Check out their World Wide Web address at httpwwwmathworkscom Software is available through their FTP site at ftpmathworkscomi 136 C319 STAJYSTYCLAL APLALYTHS function afit siga yy chisqr linregXysigma Z Function to perform linear regression fit a line Z Inputs Z X Independent variable X y Dependent variable X sigma Estimated error in y X Outputs Z afit Fit parameters a1 is intercept a2 is slope Z siga Estimated error in the parameters a Z yy Curve fit to the data X chisqr Chi squared statistic N lengthX temp sigma 2 s sumtemp sx sumx temp sy sumy temp sxy sumX y temp SXX sumX 2 temp denom ssxx sx 2 afit1 sxxsy sxsxydenom afit2 ssxy sXsydenom siga1 sqrtsXXdenom siga2 sqrtsdenom yy afit1afit2x Z Curve fit to the data chisqr sum y yysigma 2 Z Chi square return Figure 91 Garcia7s program inregm for computing the weighted least squares linear t to data using MATLAB 92 CURVE FITTING 137 and in fact any polynomial function7 is just larger values of m and successive powers of x The functions x of course7 don7t have to be power laws7 but any function you like These are called linear functions77 because they are linear in the free parameters In general7 they are not linear in x so dont get these two uses of the word linear77 confused The solution for a general linear tting problem takes the form F11 F12 Flm b1 EfiWiMi F21 F22 quot39 F2m 52 i 2 f239 where ij 21 fjxifkzi This reduces the job to solVing an m gtlt m system of linear equations All you really need to do is setup the matrix In fact7 the matrix is symmetric since ij FM These days there are plenty of computer programs that can do the matrix algebra for you Of course7 unless you have one of your own favorites7 I recommend you use MATLAB You might recall that MATLAB actually stands for MATrix LABoratory7 and it is in fact very well suited for doing all sorts of linear algebra problems7 just like this one You need to construct the matrix ij which is pretty simple to do7 and then let MATLAB solve the matrix equation using the 77 operator The function polyfit7 for example7 actually uses general matrix manipula tion to solve the linear t problem for a general n dimensional polynomial That is what the third argument is about As described in the Users Manual7 the MATLAB function call p polyfitgtlty n returns the coef cients pl of the function fI p196 Hmn l 1m pn1 which best ts the data points The call f POIYV3IP7X 138 CH 9 STATISTICAL ANALYSIS returns the function fz evaluated at the same zivalues as the data points For polynomial least squares tting where the points are not all equally weighted Garcia has provided another m le called pollsfm If you intend to write your own linear least squares tting code it would be a good idea to examine Garcia7s technique 923 Nonlinear Fitting If you want to t your data to some nonlinear function that is a function that is nonlinear in the free parameters then the problem is harder The approach is still the same namely form the X2 function and minimize it with respect to the free parameters but there are no general formulas This minimization job can of course be done numerically but when the number of free parameters gets large that can be easier said than done As you might imagine MATLAB contains the ability to do nonlinear t ting through numerical minimization The functions fmin and fmins minimize functions of one or more than one variable respectively They are pretty easy to use but be careful of the pitfalls You need to have a reasonable starting point de ned and then feed fmin or fmins the X2 function you want to mini mize You can pass arguments to the X2 function through the arguments of fmin or fmins There is lots of additional stuff in the MATLAB Optimization Toolbox which is devoted to all sorts of minimizing and maximizing prob lems This toolbox however is not part of the Student Edition of MATLAB but it is available on the ROS version of the program Another computer program that is very popular for minimizing functions in general but is almost always used to t data to some curve is called MINUIT and is available from the CERN Program Library The program is continuously updated but an older version is described in a paper by F James and M Roos Computer Physics Communications 10 1975343 You will likely come across other numerical minimization programs in anal ysis packages on just about any avor of computer Sometimes though you get lucky If you can linearize a nonlinear function then a simple rede nition of variables turns the job into something 92 CURVE FITTING 139 simple This is not unlike nding the right way to plot data so that a simple curve is what you expect See Sec 132 lt7s best to illustrate this with an example One nonlinear functional form you run into a lot is the simple exponential7 namely y A67 This is easily linearized7 just by taking the natural log7 lny ln A 7 w which can be t to a straight line Be careful7 though7 about the individual uncertainties Even if the points yi all have the same uncertainty 0y this will not be true for the straight line you are tting In this case7 the points lnyl will have uncertainty Thui Hym7 and you need to use the weighted averages when computing the free parameters in the t 924 X2 as the Goodness of Fit We7ll conclude this section on curve tting with a few more words about X2 This quantity is actually quite important in advanced statistical theory If you really want to learn more about it7 look at Taylor7s book and some other texts7 but for now just realize a simple way to use it Lets suppose you7ve taken your data and analyzed it by tting it to a straight line or perhaps some more complicated function You7ve included the individual point uncertainties in the t7 using the formulas like Eq 94 after including all the weights You graph the tted function along with the data7 and it comes close to most of the points so you gure you7ve done things correctly ls there any way you can be more con dent of the result ls there some measure of how good the t really is Maybe you need to use a function that is slightly more complicated7 and the additional terms are telling you something important about the physics7 or about your experiment Recall that if all points have the same uncertainty7 then the minimum value of X2 is identically equal to n 7 1 If the points do not have the same uncertainty7 then I said that you expect X2 to be around the same value if 140 CH 9 STATISTICAL ANALYSIS the individual points and their uncertainties are consistent with measuring a single value The same can be said for a straight line t That is7 if X2 is around n 7 27 then the t is pretty good7 and you probably don7t need to go looking around for other sources of uncertainty ln general7 ifthe quantity 922 sometimes called the reduced X2 de ned as X2 divided by the number of degrees of freedom7 is approximately unity7 then the t is good Recall that the number of degrees of freedom is de ned as the number of data points7 minus the number of free parameters7 or constraints If the uncertainties are truly random7 then you can even interpret the probability of data being given by your model See Taylor 93 Covariance and Correlations Let7s return to the discussion about Propagation of Errors77 in section 63 In particular7 we talked about how to combine the various contributions Sqm7 6 to get the net uncertainty Sq We listed a few possible choices7 namely 7 or 6g loqm ioqyj 7 or 6a i 62 Mg2 7 or 6g Something Else 12 I told you that in the one speci c case of random7 uncorrelated uncertainties7 the answer is clear and it is the third choice where we add uncertainties in quadrature We are going to go a little further now7 and look at the issue of correlated7 but still random7 uncertainties We are following the discussion as described in Taylor7s book Well just deal with qxy7 that is7 a function of two variables only Since we are working with random uncertainties only7 the best value is equal to the mean If we expand qxy about the mean values of z and y we have 1i QWmli 6Q 6g 7 AM he 96 aya y

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