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CONTENTSx13.3.3 The Lock-In Amplifier . . . . . . . . . . . . . . . . . . 24413.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24714 Experiment 7: Johnson Noise25114.1 Thermal Motion of Electrons . . . . . . . . . . . . . . . . . . . 25214.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 25514.2.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 25814.2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 26214.3 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . 26314.3.1 Analysis of Traces . . . . . . . . . . . . . . . . . . . . . 26414.3.2 Frequency Spectrum . . . . . . . . . . . . . . . . . . . 26414.3.3 Circuit Modifications . . . . . . . . . . . . . . . . . . . 26715 Experiment 8: The Faraday Effect26915.1 Magnetically Induced Optical Rotation . . . . . . . . . . . . . 27015.1.1 Electromagnetic Waves and Polarization . . . . . . . . 27015.1.2 Light Propagation in a Medium . . . . . . . . . . . . . 27515.1.3 The Faraday Effect . . . . . . . . . . . . . . . . . . . . 27615.2 Procedure and Analysis . . . . . . . . . . . . . . . . . . . . . . 27915.2.1 Polarization Calibration . . . . . . . . . . . . . . . . . 28015.2.2 Applying the Magnetic Field . . . . . . . . . . . . . . . 28115.2.3 Using the Lock-In . . . . . . . . . . . . . . . . . . . . . 284

CONTENTSxi15.3 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . 28516 Experiment 9: Nuclear Magnetic Resonance28716.1 Nuclear Magnetism and Precession . . . . . . . . . . . . . . . 28816.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 29216.2.1 Equipment Settings and Parameters . . . . . . . . . . . 29516.2.2 Procedure and Analysis . . . . . . . . . . . . . . . . . 30016.3 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . 30316.3.1 Spin Relaxation Times . . . . . . . . . . . . . . . . . . 30316.3.2 Magnetic Moments of Nuclei . . . . . . . . . . . . . . . 30317 Elementary Particle Detection30517.1 Ionizing Radiation . . . . . . . . . . . . . . . . . . . . . . . . 30617.1.1 Charged Particles . . . . . . . . . . . . . . . . . . . . . 30817.1.2 Photons and Electrons . . . . . . . . . . . . . . . . . . 31117.1.3 Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . 31517.1.4 Radiation Safety . . . . . . . . . . . . . . . . . . . . . 31617.2 Kinds of Particle Detectors . . . . . . . . . . . . . . . . . . . . 31917.2.1 Solid Angle . . . . . . . . . . . . . . . . . . . . . . . . 31917.2.2 Gaseous Ionization Detectors . . . . . . . . . . . . . . 32117.2.3 Scintillation Detectors . . . . . . . . . . . . . . . . . . 32417.3 Pulse Processing Electronics . . . . . . . . . . . . . . . . . . . 331

CONTENTSxii17.3.1 Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . 33117.3.2 Discriminators and Single Channel Analyzers . . . . . 33217.3.3 Processing Logic Signals . . . . . . . . . . . . . . . . . 33317.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33418 Experiment 10: Radioactivity33718.1 Nuclear Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 33818.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 34318.2.1 Particle Counting Statistics . . . . . . . . . . . . . . . 34418.2.2 Detecting Radiation . . . . . . . . . . . . . . . . . . . 34618.2.3 Half Life Measurements . . . . . . . . . . . . . . . . . 34819 Experiment 11: Positron Annihilation35919.1 Correlated Pairs of γ-Rays . . . . . . . . . . . . . . . . . . . . 36019.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 36219.2.1 Procedure and Analysis . . . . . . . . . . . . . . . . . 36419.3 γγ Angular Correlation in60Co . . . . . . . . . . . . . . . . . 37020 Experiment 12: The Compton Effect37520.1 Scattering Light from Electrons . . . . . . . . . . . . . . . . . 37720.1.1 Relativistic Kinematics . . . . . . . . . . . . . . . . . . 37720.1.2 Classical and Quantum Mechanical Scattering . . . . . 379

CONTENTSxiii20.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 38220.2.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 38520.2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 38720.3 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . 39120.3.1 Recoil Electron Detection . . . . . . . . . . . . . . . . 39120.3.2 Extracting the Differential Cross Section . . . . . . . . 393A Principles of Quantum Physics397A.1 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398A.2 Wavelength of a Particle . . . . . . . . . . . . . . . . . . . . . 398A.3 Transitions between Bound States . . . . . . . . . . . . . . . . 400B Principles of Statistical Mechanics403B.1 The Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 403B.2 The Maxwell Distribution . . . . . . . . . . . . . . . . . . . . 406C Principles of Mathematics411C.1 Derivatives and Integrals . . . . . . . . . . . . . . . . . . . . . 411C.2 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412C.3 Natural Logarithms . . . . . . . . . . . . . . . . . . . . . . . . 414C.4 Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . 416D A Short Guide to MATLAB419

xivCONTENTSD.1 A MATLAB Review . . . . . . . . . . . . . . . . . . . . . . . 419D.2 Making Fancy Plots in MATLAB . . . . . . . . . . . . . . . . 424D.2.1 Drea’s Handle Graphics Primer . . . . . . . . . . . . . 425

Ch 1Data Taking and PresentationProgress is made in the physical sciences through a simple process. A modelis developed, and the consequences of the model are calculated. These consequences are then compared to experimental data. If the consequences donot agree with the data, then the model is wrong, and it should be discarded.After enough successful comparisons with data, however, a model becomeswidely accepted, and progress goes on from there.Obviously, it is crucial that the data be “correct”. Furthermore, the accuracy of the measurement must also be reported so that we know how stronga comparison we can make with the model. Finally, since it is likely thatmany people will want to compare their models to the data, the experimentalresults must be reported clearly and concisely so that others can read andunderstand it.The purpose of this chapter is to give you some ideas on how to takedata “correctly”, and how to report it clearly. However, every experiment isdifferent, so these guidelines can only serve as a broad basis. You will gainexperience as you do more experiments, learning rules for yourself as you goalong.We will use some loose language, especially in this chapter. ExperimentalPhysics is a subject that can only be truly learned from experience, and termslike “settings” and “uncertainties” will become much clearer when you’ve1

CH 1. DATA TAKING AND PRESENTATION2done your time the laboratory. However, we attempt to at least roughlydefine terms as we go along. For starters, we take the term “quantity” to bethe result of some measurement, like the number read off a meter stick or avoltmeter. Things that you can change by hand, which affect the “quantity”you want to measure, are called “settings”.I will often resort to saying something like “. . . and your intuition will getbetter after some experience.” I apologize, but it is very hard to tell someonehow 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, whichcontains many good ideas about carrying out experiments: Practical Physics, G. L. Squires, Third EditionCambridge University Press (1991)1.1Your Log BookKeep a log book. Use it to record your all your activities in the lab, suchas diagrams of the apparatus, various settings, tables of measurements, andanything you may notice or realize as you go through your experiment. Thislog book will be an invaluable reference when you return to your data at anylater time, and you want to make sense of what you did in the lab. Thebook itself should have a hard binding with pages that won’t get ripped outeasily. If you make plots on graph paper or a computer, they can be attacheddirectly into the log book with tape or staples. However, a good log bookhas pages with both vertical and horizontal rulings, so that you can makehand drawn graphs directly on the page. Never write data down on scratchpaper so that you can do work with it before putting it your log book.Your log book should be kept neat, but not too neat. What’s importantis that you record things so that you can go back to them at a later time andremember details of what you did. Record your activities with the date andtime, especially when you’ve returned to recording things after a delay. Whenyou are setting up your experiment, don’t worry about writing everythingdown as you go along, but wait until things make some sense to you. That

1.2. COMMON SENSE3way, whatever you write down will make better sense when you go back toread it later.Some scientists keep a log book as a daily diary, recording not only theirmeasurements, but lecture and seminar notes and other similar things. Agood tip is to leave the first few pages blank, and fill them in with a table ofcontents as you go along. How you organize your lab book(s) is up to you,but it is probably a good idea to keep a lab book specifically for your labcourse.1.2Common SenseFor virtually any experiment, there are some good rules to keep in mindwhile you are taking data. It is a good idea to step back once in a while,during your experiment, and ask yourself if you are following these rules.In later chapters we will be more precise with language regarding “experimental uncertainty” or “measurement error”. For the time being, however,just take these terms at face value. They are supposed to indicate just howprecisely you have measured the desired quantity.1.2.1Use RedundancyIf you measure something with the various settings at certain values, youshould in principle get the same value again at some later time if all thesettings are at their original values. This should be true whether you changedthe settings in between, or if you just went out for a cup of coffee and leftthe apparatus alone. It is always a good idea to be redundant in your datataking. That is, check to make sure you can reproduce your results.In practice, of course, you will not get the same result when you comeback to the same settings. This is because any one of a number of thingswhich you did not record (like the room temperature, the proximity of yourlab partner, the phase of the moon,. . . ) will have changed and at least some

4CH 1. DATA TAKING AND PRESENTATIONof them are likely to affect your measurement in some subtle way. Withsome experience, you will be able to estimate what is or is not an acceptiblelevel of reproducibility. In any case, the degree to which you can reproduceyour results will serve as a measure of your experimental uncertainty for thatquantity.Be aware of any trends in your measurements as you take data. Youcan be redundant also by specifically taking data with settings that testany trends that you notice. If you expect data to follow a trend based onsome specific model, then take more data than is necessary to determine theparameters of the model. For example, suppose you are testing the notionthat temperature T is a linear function of pressure P , i.e.T a bPThen, the parameters a and b can in principle be determined with only twomeasurements of temperature at specific settings of the pressure. This is notredundant, however, and you should take data at more pressure settings toconfirm that the linear relation is correct. If it is not, then that tells yousomething important about either your experimental setup or your model,or both.It is natural to take data by changing the setting(s) monotonically. Thatis, to increase or decerease a setting over the range you are interested. It isa good idea to at least go back and take a couple of points over again, justto make sure things have not “drifted” while you took your data. A moreradical alternative is to take your data at more or less random values for thesettings.Don’t drive yourself crazy by changing more than one setting at a timewhile you are making measurements. Unless you are testing some trend youmay have noticed, you will certainly want to go back to find out each of thesettings affected the measured quantity.1.2.2Be Precise, But Don’t Go OverboardIt is of course important to strive for as much precision as possible in yourmeasurements. However, do not waste your time measuring one particular

1.2. COMMON SENSE5quantity very precisely if the result your are ultimately interested in, dependson some other quantity which is known much more poorly. For example,suppose you want to ultimately determine the velocity v for some objectmoving in a straight line. You do this by measuring the distance L that ittravels in a period of time t, i.e.v L/tThe relative precision of L and t contribute equally to the relative precisionof t. (We will return to this in a later chapter when we discuss experimentaluncertainties.) That is, if both L and t are both known to around 10%, theywill both contribute to the uncertainty in v. However, there is no point intrying to figure out a way to measure t, say, to 1% if you cannot measure Lto comparable precision. Your good idea for measuring t, although it may beuseful and satisfying for other reasons, will not help you determine v muchmore precisely.It is important to keep in mind how precisely you are measuring thevarious quantities that go into your final result. With experience, you willdevelop a good insight for knowing when enough is enough.1.2.3Measure RatiosWhenever you can, use your apparatus to determine ratios of quantities measured at different settings. This is a very useful technique, since commonfactors cancel when you take a ratio, and the uncertainty in these factorscannot affect the ratio. Hence, a measurement of a ratio will be inherentlymore precise than a measurement of an absolute quantity. Some of the quantites we will measure in the experiments are ratios, and they typically aredetermined with relatively high precision.Even if the ultimate goal of the experiment cannot be expressed as aratio, try to find ratios among your data that you can use to test the model.For example, suppose you want to determine the resistivity ρ of some metalsample from the decay lifetime τ of some transient voltage signal. Your modelsays that τ depends on ρ through the relationτ R2 /ρ

6CH 1. DATA TAKING AND PRESENTATIONwhere R is the radius of the sample. Even though you cannot determine τdirectly from a measurement of a ratio, you can measure τ for two differentsamples of the same metal, but with different radii R. The ratio of thelifetimes 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 bepointed out along the way as we discuss the various experiments.The determination of the lifetime of the free neutron is a good historicalexample of the triumph of ratios over absolute measurements. Free neutrons decay with a half life of around 10 minutes. Furthermore, up untilquite recently, “samples” of free neutrons were only available in fast movingstreams from nuclear reactors. Through the 1970’s, the neutron lifetime wasdetermined through two absolute measurements, one of the decay rate as thestream passed through some detectors and the other of the flux of neutrons inthe stream itself. The various measurements of different groups did not agreewith each other, and the resulting large uncertainty in the lifetime had serious consequences in astrophysics and particle physics. Then in the 1980’s,using a result based on the accepted model for neutron decay, a differentgroup measured a single ratio which agreed with previous, but less precise,measurements of this ratio and finally pointed the way to the correct value ofthe neutron lifetime. A fine account of these measurements is given in “HowLong Do Neutrons Live?”, by S.J. Freedman, in Comments in Nuclear andParticle Physics, 19(1990)209.1.2.4Avoid Personal BiasNobody starts working on an experiment without at least a rough idea ofwhat he or she is supposed to measure. It is impossible, therefore, to have nonotion of what to expect from the measurement of some quantity for somerange of settings. Sometimes, though, the result of a measurement is quitesurprising and may be an important clue to how nature works! You mustwalk a pretty fine line between what you expect and what you are trying tolearn. This will again become easier and more natural with experience.Never fudge your data to give you the answer you expect! You will notlearn anything from this, and you may miss something very important. There

1.3. TABLES AND PLOTS7are several great examples in the history of science where highly regardedresearchers 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 FirstNeutral Current Experiments Ended”, by Peter Galison, in Review of ModernPhysics, 55(1983)477.1.3Tables and PlotsA picture is worth a thousand words. It is always best to display your datausing either a table or a plot, or both. Tables are particularly useful if youwant someone else to be able to take your numbers and test a different modelwith them. Plots are best if you want to show features in your data thatmay be particularly important, such as “peaks” or “valleys” that demonstratesome phenomenon happening at a particular setting, or “trends” like linearor exponential behavior which may or may not support some specific model.It is a real art to know just how much information to include on a table orplot. Too little data can leave the reader without enough to figure out whatcan be concluded from the experiment. On the other hand, if you put toomuch on the page it is very frustrating to know exactly what the importantpoint is. As with most things in Experimental Physics, experience will bethe most important teacher.1.3.1Tables of Data and ResultsIn the old days, data was recorded directly from the instruments into the logbooks. In modern times, however, we usually use some so

and then working on your own. There is at least one book, however, which contains many good ideas about carrying out experiments: Practical Physics, G. L.