a guide for the perplexed experiments in physics (version ...slee/2402/lab/allchapters.pdf · a...

A Guide for the Perplexed

Experiments in Physics

(Version 4.0 Spring 2009)

Nural Akchurin, Mohammad Alwarawrah, Ken Carrel, Ross CarrollTexas Tech University,Department of Physics,

Lubbock, TX, USA

January 24, 2009

Contents

1 Getting Started 9

2 The Speed of Light and Sound 11

2.1 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Experimental Setup and Procedure for c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Speed of Sound in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Speed of Signal in a Coaxial Cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 The Photoelectric Effect and Planck’s Constant 17


3.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 The Franck-Hertz Experiment 21

4.1 Background Information and Performing the Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Radioactivity 25



5.2.1 Measuring the Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2.2 Measuring the Activity of Various Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2.3 Intensity versus Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2.4 Attenuation of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 The Rydberg Constant 29

3

4 CONTENTS


6.2 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.2.1 Basic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.2.2 Measurements (general) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6.2.3 Helium standard spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.2.4 The Balmer lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.3.1 Relation between wavelength and angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.3.2 Calculation of the wavelengths and the Rydberg constant . . . . . . . . . . . . . . . . . . . . . 32

7 X-ray Scattering(Bragg reflection) 33


7.2 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

8 Millikan Experiment 39


8.2 Real Millikan Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

8.2.1 Setup and Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

8.2.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

8.2.3 Calculations and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9 Physics of Gamma Spectroscopy 45

9.1 Calibration and Energy Resolution of Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

9.2 Energy Loss by Gamma Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

9.3 Deeper Look into 60Co Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

10 Physics with Cosmic Muons 53


10.2 Detection of Cosmic Muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

10.3 Cosmic Rays in Lubbock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

10.4 Energy Spectrum of Cosmic Muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

List of Figures

2.1 A detailed view of the experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Shows the equipment needed for this experiment and the way it is set up. . . . . . . . . . . . . . . . . 13

3.1 An example plot obtained by this experimental method. . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 A simple diagram of the experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1 Experimental data of the Franck-Hertz experiment for both the mercury and neon filled tubes. . . . . 22

6.1 Schematic setup for the measurement of the Rydberg constant. . . . . . . . . . . . . . . . . . . . . . . . . . 30

7.1 X-ray transitions in level schemes with or without fine structure, and the measured X-ray spectrum. . . . . . 34

7.2 Bragg reflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7.3 Experimental data showing both the first and second order Bragg peaks for cases with and without a filter. . . 36

8.1 A diagram showing the experimental setup to be used (adapted from the Oil-drop Experiment Wikipediawebpage). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9.1 The decay chain shows that 13755 Cs decays to 137

56 Ba via beta decay, n→ p+ e− + νe. The 661.66 keVgammas are produced by the subsequent decay of the excited 137

56 Ba to its ground state. . . . . . . . . 45

9.2 6027Co decays to 60

28Ni via beta decay, n→ p+ e− + νe, and 99.93% of the time to the 4+ state of 6028Ni.

The 1173 keV gammas are produced by the subsequent decay to the 2+ state, and 99.98% of the timethe 1333 keV gammas in the transition from 2+ state to the ground state. . . . . . . . . . . . . . . . . 46

9.3 Photon total cross sections as a function of energy in carbon and lead showing contributions fromdifferent processes (see text for details) [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

9.4 Top left plot is the measured spectrum of 60Co isotope with a NaI(Tl) detector in linear scale inordinate. Two photopeaks are clearly visible at 1173 and 1333 keV. Top right plot is the samespectrum but plotted in logarithmic scale where a clear third sum-peak is visible. The bottom leftcurve shows the fitted calibration curve of the form y = mx + b between the measured counts fromthe detector to energy (keV) units using the three known peaks. The bottom right plot displays theprecision of this calibration curve where the percentage difference between the fitted curve and thedata points are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5

6 LIST OF FIGURES

10.1 Vertical fluxes of cosmic rays in the atmosphere with E > 1 GeV estimated from the nucleon flux.The points show measurements of negative muons with Eµ > 1 GeV (from [9]). . . . . . . . . . . . . . 55

10.2 The angular distribution of the cosmic ray rate is symmetric around the vertical direction. The solidline is a fit of the form cos2 θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

10.3 The setup to measure the spectrum of vertical cosmic muons. . . . . . . . . . . . . . . . . . . . . . . . 57

List of Tables

9.1 Calibration measurements with γ rays. Emeas1 refers to the measured peak position of the most

dominant decay where σ(Emeas1 ) is the measure (standard deviation) of width of the peak. Eacc

1 refersto the accepted value for γ energy. The subscript 2 refers to the second most frequent decay mode. . 48

7

8 LIST OF TABLES

Chapter 1

Getting Started

Your teaching assistant will discuss the schedule and the list of experiments you will work on this semester.

Many of the experiments described in these pages are based on discussions and contributions from TTU colleagues.Walter Borst, Roger Lichti, Richard Wigmans, Ron Wilhelm and several others generously contributed to the devel-opment of these experiments. Ken Carrell has been carefully working through these experiments before we considerthem for the students. Mohammand Alwarawrah did the same in Fall 2008. Many TTU undergraduates workedon the experiments and helped improve the content: Eric Andersen, Austin Meyer, Gary Stinnett (in Spring 2007)Stephen Torrence, Dylan Smith, and Sarah Goff (in Spring 2008).

We encourage you to use analyses tools, such as Maple, MATLAB, FreeMAT, Origin, Root, etc. All of these programsare available in the computers in the lab (Sci 301). It pays to be able to code, especially C and/or C++.

9

10 CHAPTER 1. GETTING STARTED

Chapter 2

The Speed of Light and Sound

The constancy of the speed of light is fundamental to relativity theory and is a property which scientists have beenmeasuring (or attempting to do so) for a few hundred years. In this lab you will be following in the footsteps ofmany great scientists by making your own measurement of how fast light travels. In addition, you will measure thespeed of sound in solids and speed of a pulse in coaxial cables for comparison.

11

12 CHAPTER 2. THE SPEED OF LIGHT AND SOUND

2.1 Background Information

The speed of light, c, is quite a large value. For this reason people like Galileo, who attempted to directly measurec using lamps and shutters operated by lab assistants at some fixed distance apart, measured mostly the reactiontime of the assistants and not the actual transit time of the light.

The first indication that c was very fast came from astronomical observations. One of the moons of Jupiter, Io, orbitsthe planet rather rapidly and the eclipse pattern it exhibits depends on the distance between the Earth and Jupiter.Since the Earth to Jupiter distance varies over the course of a year one can use this information and the changingpattern of Io’s eclipses to determine how fast light travels. Using this technique a determination of the speed of lightwas made around 1700 and was within 25% of the correct value.

In the 19th century scientists tried to refine Galileo’s method of determining the speed of light in a laboratory settingby using complicated setups. At the current time we have electronics and other tools available to us that can beused to greatly simplify this measurement and determine c with a decent precision. Using lasers or some other fastlight source and fast photodiodes it is possible to generate and measure pulses of light with durations on the orderof nanoseconds. This means that if we can make light travel some known distance and measure the duration of thetrip, we can directly determine c.

Questions:

1. Estimate a distance that light must travel in order to have a time duration on the order of what a fast photodiodecan measure.

2.2. EXPERIMENTAL SETUP AND PROCEDURE FOR C 13

2.2 Experimental Setup and Procedure for c

Figure 2.1: A detailed view of the experimental setup.

Figure 2.2: Shows the equipment needed for this experiment and the way it is set up.

The experimental setup you will be using can be seen in figures 2.1 and 2.2. An LED (Light Emitting Diode), abeam splitter and a photodiode are all enclosed in a single box. A lens and reflector must be used to direct lightalong a certain path and an oscilloscope will be used to measure the results. Before we can perform the experimentwe must know the details of the light path we will create and the way in which we can use this setup to measure c.

An LED will produce red light that will be sent to a beam splitter (labelled S in figure 2.1). The light is split intwo with half the light going to reflector T2, getting reflected back and arriving at the photodiode used as a detector(labelled D). The other half of the light incident on S is transmitted through and gets focused into a parallel beamby the lens L. This light is reflected back by reflector T1 that has been placed at some known large distance s fromthe light source. This reflected beam travels backwards on the same path it came from and gets reflected into D byS. Since the path length of the two legs of the split beam are not the same we will see a time difference in the pulsesmeasured by the photodiode.

Setup the equipment as shown in figure 2.2 being careful to align the reflector T1 and the lens L so that light willbe reflected back into the photodiode. Use the “trigger” output from the LED/detector box as an external triggerto the oscilloscope. Once pulses are being seen on the oscilloscope, adjust the reflectors so that the amplitudes ofthe two pulses are roughly equal. Now a time difference can be measured between the two peaks. An alternative to


having both pulses shown at the same time is to block light from first one leg and then the other and measuring theposition of the peak in each case then subtracting their values to find the time difference.

Using the measured time and distance differences between the two legs split by S determine c for a number of differentdistances s.

Questions:

1. Which method of measuring the time difference (using the two peaks or blocking one leg and measuring oneat a time) is easier? Which do you think would provide more reliable results and why?

2. Determine your measured value for c both statistically (i.e. finding the mean and standard deviation of allyour measured values) and graphically (by plotting your results and determining a best fit line for your data).Include in this value your best estimates for experimental errors, describing each source of possible error.

3. How close was your measurement to the accepted value for c = 299, 792, 458 m/s? Is your measured valuecorrect within errors?

4. How would your measurements and results differ if some transparent material (like glass) was placed in thelight path between the LED/detector box and T1?

2.3. SPEED OF SOUND IN SOLIDS 15

2.3 Speed of Sound in Solids

The purpose of the experiments here is to measure the speed of sound in solids accurately and understand the reasonswhy the speed of sound is higher or lower in a given material. The experimental setup consists of a 1-m long metalrod which is connected to piezoelectric crystals at both ends. When an impulse is generated at one end, the timedifference between the two signals gives the time of travel for that sound wave. The piezoelectric crystals for thissetup are made from lead zirconate titanate (EC-64 from Edo Corporation).

Questions:

1. What do you measure the speed of sound to be in aluminum, brass and steel? What is the experimentalprecision in these measurements? Describe how you arrive at these values.

2. Sketch the two pulse traces from both crystals and explain what each pulse represents.

3. What physical parameters determine the speed of sound in a medium? Density? Electrical conductivity?Elastic modulus? Something else?

4. What do you find for the speed of sound for the mystery rod? Can you explain your result.

5. What is the effect of temperature on the speed of sound? You can use a heat gun to heat the bars to makeyour measurement.

6. Develop a classical model that describes how the sound travels in a medium that is consistent with yourmeasurements and the parameters you considered (ρ, T,B, etc).

7. Look up piezoelectric crystals and explain how they work. Explain how they generate electric pulses that wemeasure. Find out how many volts would be generated per meter for these crystals? Explain your numbers.


2.4 Speed of Signal in a Coaxial Cable

The purpose of this measurement is to determine how fast an electrical signal propagates in a coaxial cable andcompare it to c. Using a long coax cable (RG58U), a pulse generator and an oscilloscope, we can determine v. Themethod of measurement is much the same as the previous measurements of speed of light and sound in media, butin this case, we record the total round-trip travel time of the electrical pulse. The generated pulse triggers the scope,travels down the cable, gets reflected from the end and recorded by the scope.

Questions:

1. What do you measure for the speed of signal in RG58 coaxial cable? What is your precision?

2. How does this value compared to c?

3. How does your measurement compare to the accepted value?

4. What physical properties determine v? Discuss.

5. How does the pulse get reflected from the end of the cable? What is the phase of the reflected pulse? Explain.

6. There are other coax cables for you to test (RG62/U and RG59). Compare them against RG58U.

7. Calculate the capacitance and inductance per unit length for each cable. Can you calculate the characteristicimpedances?

Chapter 3

The Photoelectric Effect and Planck’sConstant

When light is incident on a metal surface, electrons bound within the metal can be released in a process knownas the photoelectric effect. This process does not follow classical physics predictions and helped lead the way to abetter understanding of the quantum world. In this lab you will use the photoelectric effect to help determine thefundamental constant related to quantum physics.

17

18 CHAPTER 3. THE PHOTOELECTRIC EFFECT AND PLANCK’S CONSTANT


The photoelectric effect has both historical and practical significance. Experimental observations of the photoelectriceffect were one of the things that led Einstein to come up with the idea of the photon and this effect is the basicprocess which makes many modern light detectors work.

The minimum amount of energy required to release an electron from the surface of a metal is known as the workfunction, φ, of the material. This energy is typically on the order of a few electronvolts. If one analyzes thisprocess from a classical physics perspective one comes to the conlusion that there is a certain minimum intensity anelectromagnetic wave would need to transfer enough energy to release electrons from a metal. One would also expectthat the energy of an electron released from the metal by this process would depend on the intensity of the light waveand not the frequency of the light. The experimental results, however, proved this line of thinking to be false andinstead found the following set of rules that govern this process. First, in order for a photoelectron to be emitted,the frequency of the incident light must be higher than a certain cutoff value, regardless of the intensity of the light.Second, the number of photoelectrons that get emitted per second (known as the photocurrent) is proportional tothe intensity of the light. And lastly, the energy of the emitted photoelectrons increases with the frequency of theincident light.

To adequately describe what was happening, Einstein made the assumption that light interacts with the electronsfound in the metal just as a stream of particles would. These “particles” of light he named photons and possessed awell-defined energy given by E = hf where f is the frequency of the light and h is a fundamental constant known asPlanck’s constant. In this case the maximum amount of kinetic energy a photoelectron could obtain would be thedifference between the energy of the light and the work function of the material:

Emax =h

ef − φ (3.1)

Using this equation and experimentally measuring Emax will allow us to measure the ratio of constants in front of f .

Questions:

1. Is the photocurrent dependent on the frequency of light? In other words, does the photocurrent depend on theenergy of the photoelectrons? Why or why not?

2. Visible light is capable of producing the photoelectric effect. What is the energy of photons in this frequencyrange?

3.2. EXPERIMENTAL PROCEDURE 19

3.2 Experimental Procedure

For this experiment you will be using a mercury lamp, a monochromator and a photocathode inside a box withan ammeter on top. The first thing that needs to be done is to find the three spectral lines of the mercury lamp.To do this, align the mercury lamp and monochromator so that light from the lamp passes through the slits of themonochromator. Open the slits of the monochromator so that you can easily see light passing through and changethe wavelength of light allowed to pass through the monochromator until you find the three very bright lines. Recordthese values.

Plug in and turn on the box with the photocathode. Make sure to cover the opening and allow the box to sit for a fewminutes. After this, use the appropriate knob to adjust the ammeter so that it reads exactly zero. Using the providedstands align the three pieces of equipment so that light passes from the lamp through the monochromator and ontothe photocathode. You will be recording values for the photocurrent using the ammeter on the box enclosing thephotocathode. Values for the opposing voltage can be found by using the connectors on the photocathode enclosureand a multimeter.

You will perform the experiment in two ways and judge which method is the best/easiest to perform. The firstmethod will be to adjust the slits on the monochromator so that each of the three spectral lines have the same initialphotocurrent (i.e. when there is no opposing voltage). From there, adjust the voltage so that the ammeter is on aparticular current and record the current and voltage. Make measurements for several photocurrent values all theway down to zero, which is known as the stopping potential. The second method will be similar, but instead ofadjusting the slit for each spectral line you will set the slit width using the blue line and use this same width for theother two spectral lines.

Taking the stopping potential for each spectral line as Emax in equation 3.1 and the wavelengths you found for thespectral lines of the mercury lamp, graph your results and determine both the ratio h/e and φ. You can expectsomething that looks similar to figure 3.1 when you plot current versus voltage for each of the three spectral lines.

Figure 3.1: An example plot obtained by this experimental method.

Questions:

1. Describe what you think is happening inside the photocathode using physical arguements. In doing this, nameeach of the 7 key components to this experiment labelled ’A’ through ’G’ in figure 3.2. Hint: How is a currentproduced and what is the role of the opposing voltage?

2. Compare the relative intensities of the emission lines of the mercury lamp. Hint: You can determine this from

20 CHAPTER 3. THE PHOTOELECTRIC EFFECT AND PLANCK’S CONSTANT

Figure 3.2: A simple diagram of the experiment.

the second experimental method. Do these relative intensities agree with the color of light you see coming fromthe lamp?

3. Why is the stopping potential the same as Emax?

4. How close is your experimental measurement of h/e to the actual value?

5. What are possible sources of errors? Provide an estimate of your errors to include with your experimentalvalue.

6. Which experimental method was easiest to perform? Which was the most accurate?

7. Is there some physical reason why one of the methods would be better or worse than the other? If so, explainhow.

8. Could this experiment be performed with a light source that did not have discrete emission lines? Why or whynot?

Chapter 4

The Franck-Hertz Experiment

In this lab you will perform an experiment that was instrumental to forming basic concepts in quantum mechanics.James Franck and Gustav Hertz did the experiment in 1914 using essentially the same experimental setup and wereawarded the Nobel Prize in Physics for it in 1925. At the turn of the 20th century, physics of the small scale wasstill a very new and poorly understood field. There were two competing theories on atomic structure: Rutherford’sTheory and Bohr’s Theory. The theories were similar but differed in a few key aspects. This experiment providedclear evidence in support of one of these two theories.

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22 CHAPTER 4. THE FRANCK-HERTZ EXPERIMENT

4.1 Background Information and Performing the Experiment

When current flows through an electrical circuit, Ohm’s Law gives us the exact relation between the voltage (V) andthe current (I):

V = IR

If we replace the usual current carrying device - a copper wire - with something else do we expect the same relationto hold? In other words, does Ohm’s Law apply to current flowing only in a copper wire?

The main piece of experimental equipment we will use is a gas filled tube. If you look at one of the tubes you willnotice three things inside it. There is a cathode, a grid anode and a collector electrode. The grid anode is keptat a positive voltage with respect to the cathode so that electrons are accelerated toward the grid. The collectorelectrode is kept at a slightly negative voltage so that only electrons with a certain energy threshold can reach it. Onthe front of the heating oven for the mercury tube is a wiring diagram and connectors that are labelled so wiring tothe power supply can be done easily. When connected to the tubes, the power supply will provide a current throughthe tubes by accelerating electrons through a certain voltage. It measures the current by determining how manyelectrons make it through the gas to the collector electrode. This allows us to determine how voltage and currentare related when electrons flow through the gas in the tubes.

We will first do the experiment with the neon filled tube. Make all the connections (double check to be sure youhave it wired correctly and ask if you have questions - this equipment can be damaged if powered on with a wrongconnection) and allow the tube to warm up before taking any measurements. Set the switch in the middle of thepower supply to ’Ramp 50 Hz’ and adjust the available knobs to get the best curve possible on the oscilloscope.Make a sketch of the graph you see (it should look similar to figure 4.1) and make quantitative measurements of thevarious features (take note of the scaling of the output from the power supply).

Figure 4.1: Experimental data of the Franck-Hertz experiment for both the mercury and neon filled tubes.

Once becoming familiar with the setup and experiment you will make more precise measurements using a multimeter(the switch in the middle of the power supply must be set to ’Man.’ for this part). Find the locations of as manymaxima and minima as possible. This can be accomplished in one of two ways: 1) connect multimeters to both the

4.1. BACKGROUND INFORMATION AND PERFORMING THE EXPERIMENT 23

voltage and current outputs and find the largest (smallest) current to determine the local maximum (minimum) or2) watch the oscilloscope to find the local maximum or minimum. As you adjust the voltage on the power supplywatch the tube through the window. There should be small areas of light visible: note their position, intensity andsize as you adjust the voltage. After determining the voltages of the maxima and minima of the current, use theprovided spectrograph to determine what wavelengths of light are coming from the tube.

Repeat the above procedures and measurements for the mercury filled tube. The mercury tube must be heatedso that the vapor inside can be at the right pressure for the phenomenon we are after to be seen. Adjust thetemperature (between ∼150 and ∼200 degrees) as well as the knobs on the power supply to obtain the best curveon the oscilloscope.

IMPORTANT: Both tubes are fragile and can be damaged easily. Also, the mercury tube gets very hot so do notallow anything to come into contact with the enclosure.

Questions:

1. Does the current flowing through the gas filled tubes obey Ohm’s Law? What would the trace on the oscilloscopelook like if it did?

2. Are there differences between how current flows through a copper wire and a gas? If so, what do you think isdifferent?

3. Briefly explain the theories proposed by Rutherford and Bohr to describe atomic structure. In what ways arethe two theories different?

4. Describe the graph you obtained and try to explain its features using physical arguments (it may be useful toinclude how the adjustments made with the knobs affected the graph and how the features of the visible lightemitted from the neon tube changed with increasing voltage).

5. Which of the two theories on atomic structure does this experiment help to prove is correct? Why?

6. Using the values for the maxima and minima for each tube determine the lowest excitation energy for mercuryand neon. How do your results compare with the accepted values of ∼4.9 and ∼19 eV respectively?

7. Can you account for these energies with the light emitted from the tube(s)? Why or why not?

24 CHAPTER 4. THE FRANCK-HERTZ EXPERIMENT

Chapter 5

Radioactivity

At the end of the 19th century and into the beginning of the 20th century many important and fascinating discoverieswere made in the physics of the very small scale. One of the discoveries that has had possibly the greatest impactwas radioactivity. Numerous wonderful (and a few regrettable) applications have been found for these remarkablediscoveries.

25

26 CHAPTER 5. RADIOACTIVITY


The credit for discovering radioactivity is usually attributed to Henri Becquerel who found by accident that uraniumexposes photographic plates. Immediately following this discovery, work by Marie and Pierre Curie as well as ErnestRutherford showed that radioactivity was much more complicated than had first been thought and that new physicswould be needed to describe how it worked.

Although it was not known at the time of the discovery of radioactivity, each atomic element can have numerousdifferent configurations inside the nucleus. Elements are determined by the number of protons found inside thenucleus, but the number of neutrons can vary. These atoms of the same element that differ in the number ofneutrons are called isotopes. There are several forces at work inside a nucleus so it comes as no surprise that someconfigurations of protons and neutrons are not stable. When this instability occurs, the system naturally falls intoa stable configuration and this is accomplished through radioactive decay. A single element may have one or manystable isotopes and can similarly have a varying number of radioactive isotopes.

In order for an unstable nucleus of an atom to become stable, it must lose energy by emitting what is known asradiation. The radiation is either some particle (or particles) or is an electromagnetic wave. At the atomic level thisprocess is completely random; that is, one can not predict whether a particular atom will decay or not. On average,however, this phenomenom does follow a predictable behavior. Given some number N of a radioactive isotope, weexpect that the number of decay events dN in some time dt should be proportional to the number of isotopes present.So,

−dN = Nλdt

where λ is some proportionality constant. The above is simply a first-order differential equation whose solution isknown to be:

N(t) = N◦e−λt (5.1)

In this equation, N◦ is the original number of isotopes and N(t) is the number after some time t. The λ in this caseis the decay constant of the exponential decay and is usually replaced by either the mean lifetime, τ :

τ = 1λ

or by the half-life, given by t1/2, which is defined as the amount of time for half of the radioactive nuclei to decay:

t1/2 =ln 2λ

= τ ln 2 (5.2)

The half-life of a radioactive isotope has very important consequences and can be used to distinguish differentradioactive sources from one another in some cases.

In the following experiment you will explore the relationship given by 5.1 as well as determine the half-life of differentradioactive sources. You will also examine how radiation from a radioactive source interacts with different materials.The measuring device you will be using is a Geiger counter which has a probe connected to a meter that readsradioactivity rates in disintegrations per minute (or counts per minute - CPM) or something similar. Other typicalunits used to describe radioactivity are becquerel (Bq) which is the number of disintegrations per second and curie(Ci) which is 3.7 × 1010 Bq.

5.2. EXPERIMENTAL PROCEDURE 27


5.2.1 Measuring the Background

Anywhere you are on the Earth there is always some amount of radiation that passes through a given area in agiven amount of time. This is known as background radiation. Typical sources of this radiation include cosmic raysand naturally radioactive materials found in building materials. You first want to find how much this backgroundis in the area you are going to perform the rest of your experiment. Turn on the Geiger counter and count the ratefor some short amount of time (you will probably have to count the actual audible clicks of the counter instead ofreading the meter). Using the result from your short measurement, estimate how long you will have to count in orderto get a rate that is accurate to ∼5% and then count for that period of time to get your final background value. Thisvalue (with it’s uncertainty) will need to be subtracted from all of your other measurements as a known backgroundto your measured signal.

Questions:

1. Determine the background radiation rate for several different probe orientations and at different locationswithin the room. Are any of your results significantly different? If they are different, suggest a reason why thisis so.

5.2.2 Measuring the Activity of Various Sources

You will now measure how active each of the individual sources are. Place each source up to the probe and eithertake the measurement from the meter or count the audible clicks as you did for the background measurement. Besure each of the sources are placed in the same spot on the probe to help eliminate any systematic error.

Questions:

1. Does it matter which side of a particular source faces the probe? Take measurements for one of the sources indifferent orientations with respect to the probe to see if there is any difference. Explain the results.

2. Calculate the total activity for each source. Hint: Think about how the previous question applies to thisquestion and be sure to subtract the background rate.

3. Using your calculated activity, the activity printed on the source and the date stamped on the source, determinethe half-life for each of the radioactive sources using equations 5.1 and 5.2. How close are your calculated valuesto the ones listed on each of the sources? (Be careful to estimate your uncertainties and think of any systematicuncertainties you may have.)

4. Based solely on the activity (and calculated half-life), what is the unknown radioactive source likely to be?

5. Make a plot of activity versus time for each of the sources and include the function that should describe theradioactive process. How well do the points agree with the function?

6. For which of the radioactive sources should you repeat the measurement in a few days if you want to see anappreciable difference in the activity?

5.2.3 Intensity versus Distance

We now want to investigate how the distance from a source effects the amount of radiation you detect. Chooseone of the sources that has a large measured rate from the last section. Starting with the source against the probe

28 CHAPTER 5. RADIOACTIVITY

and moving it away, take measurements for the activity at several different distances. If you found a difference inmeasured activity for different orientations of the source be sure to carefully align the source at each distance (thisis probably a good idea even if there isn’t a difference just to be consistent).

Questions:

1. Plot your results.

2. What function seems to describe the trend you see the best?

3. What reason(s) can you think of that would cause such a relation?

5.2.4 Attenuation of Radiation

If we put some material between the radioactive source and the Geiger counter, the radioactive particles emittedfrom the source must pass through this material to be detected. This allows for some of the particles to interactwithin the material and possibly not make it out the other side to be detected. The interaction in different materialsis very important when we want to consider how best to isolate a radioactive source so that the radiation does notcause harm to anyone. For each of the measurements you make, set one of the more highly active sources a fixeddistance from the probe and then place different materials between the source and the probe without changing thedistance between the two. Using several different materials (aluminum, lead, paper, etc.) take measurements usingdifferent thicknesses of the chosen material.

Questions:

1. Make a plot of the normalized intensity (the intensity with nothing between the source and probe dividedby the intensity with something between the source and probe) versus thickness of material for each of thematerials and sources you used.

2. What relation best describes your measurements (i.e. which function best fits the data)?

3. Are there any significant differences between any of the materials and/or sources? Can you come up with areason for the differences?

5.2.5 Conclusion

Here are a few more questions to bring everything together.

Questions:

1. Was the background radiation rate significant for any of your measurements?

2. What units did you choose to use and why?

3. What is the best way to isolate radioactive sources so that they do not harm anyone (for cases in which thereis little space available and a lot of space available)?

4. Did any of the sources emit a different type of radiation than the rest? Justify your answer.

Chapter 6

The Rydberg Constant


The constant R in the formula1λ

= R (1n2

2

− 1n2

1

) (n1 > n2) (6.1)

which expresses the wavelengths of the electronic transitions in the hydrogen atom is one of the most accuratelydetermined quanties in science. Its value is known to better than 10−9, one part in a billion. In this lab, you willmeasure R (although with somewhat less accuracy than the state of the art.

Although Balmer derived an empirical formula for the wavelengths of the Balmer series (hence the name) anddetermined R in that way, it was not until the quantum theory of Niels Bohr (1913) that this constant could becalculated in terms of other measured quantities. Physically, R is related to the energy required to ionize a hydrogenatom (n1 =∞, n2 = 1→ R = 1/λ in eq. 6.1)

In this experiment, you will use a spectrometer with a high-quality reflection diffraction grating to measure thewavelengths of three of the Balmer lines of hydrogen. These measurements will be made more accurate using atechnique that is quite common in spectroscopy: comparison of the unknown wavelengths (hydrogen in this case) tothe well-known wavelengths of a standard spectrum (helium in this case). In modern laser spectroscopy, the standardspectral lines are often provided by uranium, molecular iodine or molecular tellurium, all of whose visible lines havewavelengths known to an accuracy of better than 0.0001 nm.

Once you have determined the wavelengths of the Balmer lines, you will fit the obtained values to eq. 6.1 to determineR.

6.2 Procedures

6.2.1 Basic setup

Figure 6.1 shows the geometry of the spectrometer with a transmission grating.

1. We must first determine the spacing of the grooves on the grating(s). To do this use a laser to produce thediffraction pattern on one of the walls of the room. Now measure both the distance from the grating to the

29

30 CHAPTER 6. THE RYDBERG CONSTANT

Figure 6.1: Schematic setup for the measurement of the Rydberg constant.

wall and the distance between the bright spots in the pattern on the wall. Using these you can determine theangle for different diffraction orders and use equation 6.2 to determine d (since the wavelength is known forthe laser).

2. To set up the spectrometer, begin by directing the viewing telescope at a distant object and focusing on it.

3. Then set up the helium lamp in front of the slit of the collimating telescope, so that its light passes throughthe slit. Adjust the lamp position by staring directly into the collimating scope lens and moving the lamp soas to get the most light through.

4. To focus the collimating telescope, point the viewing scope directly into the collimating scope, so that you seean image of the slit. Adjust the focus of the collimating scope until the image of the slit is sharp. Rotate theslit holder so that the slit image is vertical.

5. Record the angle (θS) reading for which the center of the slit image lies on the vertical crosshair.Note: the angle readout is in degrees and minutes of arc (1 minute of arc equals 1/60th of one degree). Youwill need to convert all of your results in decimal degrees before analyzing them!

6. Adjust the height of the grating table so that its surface is at the same height as the bottoms of the lenses ofthe telescopes.

7. Now place the grating on the table.Note: The surface of the grating is very fragile and impossible to clean. Do not touch the surface of thegrating.

8. Determine the plane of diffraction of the grating by observing the reflection of the room lights from its face.

9. Set the grating in the holder on the turntable so that it will diffract light in a horizontal plane. Do not movethe turntable during the rest of the experiment.

6.2.2 Measurements (general)

The grating equation isd sin θd = nλ (6.2)

where d is the spacing of the grating grooves, n is an integer indicating the diffraction order, θd is the diffractionangle (also measured with respect to the normal of the grating), and λ is the wavelength of the diffracted light. Forthe value of d of our grating and the geometry in which this grating is used, n equals 1 for all the lines you willobserve in this experiment.

6.3. DATA ANALYSIS 31

The measuring scale on the table may be rotated so that it may be more easily read. Set this window before takingany measurements and do not move it after. When recording the angles of diffraction be sure to take into accountthe initial angle (i.e. the angle given when the telescope is centered on the slit, θS).

6.2.3 Helium standard spectrum

There are five bright lines in the helium spectrum that we will use for standards. Two are blue, one is blue-green,one is green and one is orange. Their wavelengths are listed in the following table.

Helium line Wavelength (nm)Purple 447.15Blue 471.31

Green 1 492.19Green 2 501.57Orange 587.56

Measure the angle (θλ) of the diffracted light of each of these lines as accurately as you can. Make good use of thevertical crosshair. It may be necessary to slightly refocus the viewing oscilloscope to see the images of the slit asclearly as possible.

6.2.4 The Balmer lines

With the helium lamp still in the same position, move the viewing telescope over to view the slit and carefully centerthe crosshair on the image of the slit. Now remove the helium lamp and replace it with the hydrogen lamp.

Adjust the position of the hydrogen lamp until the image of the slit is centered accurately on the crosshair. Thisprocedure ensures that the hydrogen lamp is at the same location that the helium lamp was during the measurementsof the helium reference lines.

Now carefully measure the angles (θλ) for each of the three visible Balmer lines: The red Balmer-α line, the blue-green Balmer-β line and the blue Balmer-γ line. Beware of the many molecular hydrogen lines coming from the lamp.These are usually considerably dimmer than the mentioned atomic lines.

6.3 Data Analysis

6.3.1 Relation between wavelength and angle

Organize the data in columns. In the first column, enter the known wavelength values of the five helium lines, andin the second column the corresponding measured values for sin (θλ). Fit the wavelength column to the followingformula, which describes a second order polynomial:

λ = a+ b · x+ c · x2 (6.3)

in which x represents sin (θλ). The first two terms in this expression correspond to the grating equation (6.2), inwhich θd = θλ. The additional, quadratic term is needed to compensate for imperfect placement of the grating andother possible instrumental effects.

32 CHAPTER 6. THE RYDBERG CONSTANT

Record the fitted values of a, b and c, along with their standard deviations. Plot your fitted curve, together with themeasured data points and print it out.

6.3.2 Calculation of the wavelengths and the Rydberg constant

Next, we use the measured values of the Balmer wavelengths to calculate the Rydberg constant. Organize yourmeasured data in columns. The first column contains your measured values of sin (θλ) for the Balmer lines. Usingyour calculated values of a, b and c, you can now calculate the wavelengths of these lines with formula 6.3. Recallthe Balmer formula

1λ

= R (122− 1n2

) (6.4)

where n = 3 corresponds to Balmer-α, n = 4 to Balmer-β and n = 5 to Balmer-γ. Enter the values of the appropriateintegers, n, in column 2, and the the values of 1/n2 in column 3. In column 4, you put the values of 1/λ (i.e. theinverse of the values in column 1). Fit the columns containing 1/n2 and 1/λ to

y = a′ + b′x (6.5)

where y stands for 1/λ and x for 1/n2. Record the values of a′ and b′ and their standard deviations. Also record thecalculated wavelengths of the Balmer lines. Plot the fitted curve and print this plot out.

Comparison of the equations 6.4 and 6.5 shows that a′ corresponds to R/4 and that b′ = −R.

Questions:

1. Calculate R from a′ and b′, along with their associated uncertainties (from the standard deviations). Doyour two values agree with each other, and with the accepted value (R = 1.0968 · 107 m−1), to within theexperimental uncertainties? Which experimental value is more accurate?

2. One can also determine R by fitting the data using only a linear term with no constant and plotting 1/λ versus(1/22−1/n2). Do so and compare your result with the previous method and the accepted value. Which methodof determining R is the best in your opinion and why?

3. From the results of your calibration fit (eq. 6.3), you can calculate the separation of the grooves of the grating(d), in two different ways. Do so, and calculate the associated uncertainties. Calculate the density of groovesfor the grating, and compare to the nominal value printed on the grating (if there is one).

4. Speculate about the main sources of experimental error in this experiment, and about ways to avoid them.

5. The actual values of the wavelengths of the three Balmer lines are 434.05 nm, 486.13 nm and 656.28 nm. Whichof your calculated wavelengths differs most from the actual value? Why do you think that is?

Chapter 7

X-ray Scattering(Bragg reflection)


In 1895, X-rays were discovered by Wilhelm Rontgen. At the time of their discovery, these rays were mysterious(hence their name) and their origin was not at all understood. Now we know that one important mechanism of X-rayproduction involves the transition of electrons between different orbits in atoms.

Normally, the absorption and emission of photons in materials is carried out by electrons residing in the outer orbitsof the atoms. However, a sufficiently violent disturbance can also excite or even remove electrons from the innershells. If such an electron is removed, the atom is left with a vacancy in a low-lying excited state. In that case, anelectron from a higher orbital may “jump” to this vacant position and emit a photon in the process. The emittedphoton carries an energy equal to the difference between the energy of the atom in its initial state (with a vacancy inthe inner shell) and the final state (with this vacancy filled and a vacancy in one of the higher orbitals). Especially,for high-Z materials, this energy difference may be very high (tens of keV’s) and the emitted photon, which is anX-ray, may have a correspondingly short wavelength.

The X-rays which we will use in this experiment come from a copper target. Inner electrons from the copper atomswill be removed by bombarding this target with a stream of highly energetic electrons generated in a strong electricfield. Once an inner electron has been removed, a variety of transitions may occur and cause the atom to relax backto its original ground state.

Figure 7.1 shows some of these possible transitions, along with the characteristic spectrum (intensity vs. wavelength)for X-ray production in this setup. This spectrum exhibits two sharp peaks superimposed on a continuous background.This background is caused by bremsstrahlumg, emitted when the energetic beam electrons are decelerated in the targetmaterial. The peaks correspond to electronic transitions between inner orbitals of copper atoms as discussed above.

The level schemes in figure 7.1 show that each of the main energy states (n = 1, 2, 3, ...) is split into closely spacedsublevels. This phenomenon, called fine structure, is a result of magnetic interactions. However, the energy differencesbetween those sublevels are too small to be resolved with the apparatus for this experiment and the effects leadingto this level splitting are also irrelevant for our present purpose. Therefore, we will neglect the fine structure.

Furthermore, the probability for X-ray transitions to the n = 1 final state (the so–called K lines) is much larger thanfor those with n = 2 (the L lines) or n = 3 (the M lines). Subsequently, the K lines are the only ones intense enough

33

34CHAPTER 7. X-RAY SCATTERING (BRAGG REFLECTION)

Figure 7.1: X-ray transitions in level schemes with or without fine structure, and the measured X-ray spectrum.

to detect with our equipment.

The accurate determination of the wavelengths of X-rays is not an easy task. It has been discovered, though, that acrystal lattice with a regular atomic pattern can be utilized as a type of diffraction grating for X-rays. The methodwe will use was proposed by W.L. Bragg in 1912: X-rays incident at a certain angle θ are scattered by parallel planesof atoms inside the crystal.

Two conditions have to be met in order to establish Bragg reflection (see fig. 7.2):

1. The angle of incidence must be equal to the angle of reflection

2. Reflections from successive layers must combine constructively (constructive interference)

We will use a salt crystal (NaCl) which has a very simple structure, known as face-centered cubic. For constructiveinterference, the path length difference between radiation reflected from successive layers must be an integral multipleof the wavelength of the X-rays. Simple geometric arguments translate this condition into

nλ = AB +BC = 2d sin θ (7.1)

This equation is known as Bragg’s law.

In order to use this equation for determining wavelengths, we must first know the atomic spacing, d. For any face-centered cubic structure, this spacing can be calculated on the basis of the density (ρ) and the molecular weight (M)of the material. The mass of a single molecule is M/NA, where NA is Avogadro’s number (6.02 · 1023). Therefore,the number of molecules per unit volume is ρ/(M/NA). Since NaCl is a diatomic molecule, the number of atoms per

7.2. PROCEDURES 35

Figure 7.2: Bragg reflection.

unit volume is 2ρ/(M/NA). Therefore, the distance between adjacent atoms in the crystal structure is given by

d3 =1

2ρNA/Mor d = 3

√M

2ρNA(7.2)

Bragg reflection is an extremely useful phenomenon in the field of X-ray spectroscopy. It allows separation of X-rayswith different wavelengths much like a diffraction grating inside a monochromator separates the various colors ofvisible light. The beam of X-rays generated in our apparatus contains a whole variety of different wavelengths (see thespectrum in fig. 7.1). However, when this beam is incident at an angle θ to the crystal planes, only the wavelengthsthat satisfy eq. 7.1 will be reflected.

The wavelength of the first order reflection is found by letting n = 1, the second order corresponds to n = 2, andso on. By varying the angle of incidence and measuring the intensity of the Bragg reflection, we can determine thespectrum of the X-rays generated in the copper target. The purpose of this lab is to measure the wavelengths of thecopper Kα and Kβ lines.

In order to scan for all possible diffraction lines, the crystal (and thus the angle of incidence of the X-rays) shouldbe rotated from 0◦ to 90◦. Since the source is in a fixed position, the detector thus has to rotate from 0◦ to 180◦

to maintain the condition that the angle of reflection has to be equal to the angle of incidence. In practice, a rangefrom 10◦ to 120◦ will turn out to be sufficient. An example plot of data with and without a filter can be found infigure 7.3.

7.2 Procedures

• Mount the NaCl crystal in the crystal post ensuring that the major face having “flat matte” appearance is inthe reflecting position.

• Place the primary beam collimator in the BASIC port with the 1 mm slot vertical.

• Mount the 3 mm slide collimator at E.S. 13 and the 1 mm collimator at E.S. 18.

• Zero-set and lock the slave plate and the carriage arm cursor as precisely as possible.

• Look through the collimating slits and check that the incoming beam direction lies in the surface of the crystal.


Figure 7.3: Experimental data showing both the first and second order Bragg peaks for cases with and without a filter.

• Mount the Geiger–Muller tube and its holder at E.S. 26.

• Switch the High Voltage to 20 kV.

• Track the carriage arm around from its minimum setting (2θ = 11◦) to its maximum (2θ = 124◦). Record theintensity with the rate meter.

The carriage arm should be indexed to 2θ = 15◦ and the thumb wheel set to zero. When the scatter shield is closed,settings from 11◦ to 19◦ can be achieved using only thumb wheel indications.

Where the count rate appears to peak, plot intervals of only 10 arcminutes using the thumb wheel. At each peak,measure and record the count rate and the angle as precisely as possible.

Repeat the measurement procedure for a high voltage setting of 30 kV and for each of the other available crystals(Alum - KAlSO4, Rutile - TiO2, Calcite - CaCO3, and the ’unknown’ crystal).

Questions:

1. Determine the atomic spacing d for the NaCl crystal. The atomic numbers for sodium and chlorine are 23 and35, respectively, and the density of NaCl is 2.165 g/cm3.

2. Make a plot of intensity versus 2θ for both accelerating voltages. Superimpose the graphs. Determine whichpeaks differ only in amplitude and not in angle. For these peaks, determine θ. Let the order of reflection beone (n = 1) and calculate the wavelengths of all these peaks.

3. Let n = 2 and repeat the calculation. If any wavelength from the n = 1 case corresponds to a wavelength fromthe n = 2 case, then this tells you that you have observed a first and second order Bragg reflection for thisparticular wavelength.

4. Repeat the procedure for the n = 3 case, and see if you observe any third order Bragg reflections.

7.2. PROCEDURES 37

5. Look up the wavelengths (energies) for the Kα and Kβ lines of copper. Which of your peaks correspond tothese lines? Are the observed wavelengths (energies) within the experimental uncertainties equal to the knownvalues?

6. Calculate the wavelengths of the most energetic X-rays (λcutoff) created by the beam of bombarding electrons,for both accelerating voltages (20 kV and 30 kV)

7. Using your measured values for Kα and Kβ determine the crystal lattice spacing for the KAlSO4, TiO2, CaCO3,and unknown crystals. Look up the accepted lattice spacings for the known crytals and try to determine whatthe unknown crystal is. Are the observed spacings within the experimental uncertainties equal to the knownspacings?

8. Does the orientation of any of the crystals matter in this experiment? Explain why or why not.

Chapter 8

Millikan Experiment

Robert Millikan experimentally determined that electric charge is quantized or it came in lumps of a certain minimumsize. This minimum size is the charge on a single electron.

Millikan did this by finding the electric field required to levitate tiny oil droplets. From this field, and the mass ofthe droplets, he found the total charge on the droplets. After measuring many droplets, he determined that the totalcharge on any droplet was always some multiple of 1.6× 10−19 C. This is the charge on a single electron and as faras we know, it is the smallest unit of charge available under normal circumstances.

39

40 CHAPTER 8. MILLIKAN EXPERIMENT


To get the basic idea and before we run a detailed and complicated experiment, we want to do something simpler.Instead of measuring the charge of an electron, we will be using a similar procedure to find the mass of an unknownsmall object (let’s call it USO). There are unknown number of USOs inside sealed containers. Just as Millikan hadno way of knowing ahead of time the number of electrons on an oil drop, so you will not know the number of USOs ina given container. By measuring the mass of enough containers, you can determine the most likely mass of a singleUSO. The exact method you use for this determination is up to you but you are not allowed to open the containers.

Once you determined the method you would like to use, make your measurements and perform your analyses whichshould end up giving you mUSO ±∆mUSO. Write a short summary report (no more than 2 pages) which includesan analysis of the uncertainty of your results. Obviously, you may consult references in error analyses.

8.2. REAL MILLIKAN EXPERIMENT 41

8.2 Real Millikan Experiment

8.2.1 Setup and Technique

The real experiment that was performed by Millikan (and that you will be doing) is exactly analogous to what wasdone in the previous section. In this case, though, the containers will be replaced with drops of oil and the USOswill be replaced with electrons. The measurements will be done in a different way but the analysis will be identicalto the method you developed to find the mass of the USOs.

A schematic of the apparatus used in this experiment is shown in figure 8.1. An atomizer (spray bottle) will beused to inject small drops of oil between two parallel plates that can have a voltage applied between them. A lightsource illuminates the area between the plates and a telescope is positioned so that one can see the drops movingbetween the plates. (NOTE: The image when looking through the telescope is inverted so if a drop is moving up inthe eyepiece it is actually moving down.) The first thing that must be done is to figure out the spacing between theparallel plates and using this determine what distance each line in the eyepiece corresponds to.

Figure 8.1: A diagram showing the experimental setup to be used (adapted from the Oil-drop Experiment Wikipediawebpage).

Now take a little while to get used to spraying drops into the apparatus and observing their motion with the electricfield on and off. Selecting a single drop from a large group can be accomplished by varying the field so that all theother drops reach one of the plates. There are two measurements that you will make: 1) the velocity of a drop withno field and 2) the velocity of the same drop in the presence of an electric field. To find the velocity you must timehow long it takes for the drop to go between two lines in the eyepiece.

8.2.2 Procedure

First record the temperature and barometric pressure in the room you are performing the experiment in. Select asingle voltage to apply to the parallel plates for all of your measurements (between ∼100 and 500 volts) and recordthis value as measured on the voltmeter on the front of the apparatus. Next you want to isolate a single drop insidethe apparatus to perform detailed measurements on. Record several (∼10) velocities for this drop with and withoutthe electric field applied. Find the average velocities and calculate the charge on this drop before measuring anymore. You want to measure drops that only have one or a few charges on them. So, based on your measurement ofthe first drop, adjust the types of drops you select accordingly (i.e. pick faster or slower moving drops if need be).

Tips/Tricks:

• Practice spraying oil on a napkin before spraying it in the apparatus. You want a fine spray with not muchvolume - short hard squeezes should work better than slower ones.


• If no drops are going in the apparatus check to make sure the hole in the top plate is not clogged with oil. Ifit is, blow out the excess oil and try again. IMPORTANT: Make sure the HV is off and disconnected beforetaking the apparatus apart.

• The easiest and most accurate way of timing the motion is to start the timer when the drop passes behind oneline on the eyepiece and stopping it when it passes another.

• You want to make several measurements on the same drop so it is probably easier to have someone record thetimes you measure so you can keep an eye on the drop and keep it between the plates.

8.2.3 Calculations and Analysis

A drop of oil falling between the parallel plates with no electric field present will experience two forces - the force ofgravity pulling the drop down and a drag (friction) force. If there is an electric field present the gravitational anddrag forces will still be present but there will also be a force on the drop from the electric field. If all these forcesare equal the drop will move at a constant speed known as its terminal velocity. For the drops in our experiment,terminal velocity is reached very quickly so when measuring the speed of the drops with or without an electric fieldpresent you are measuring their terminal velocity.

By examining the free body diagrams for both cases one determines the following relations:

mg = kv1 (8.1)

Eq = mg + kv2 (8.2)

where m is the mass of the drop, g is the acceleration due to gravity, k is a coefficient of friction, E is the electricfield intensity, q is the charge on the drop and v1 and v2 are the terminal velocities for no field and with a fieldrespectively. We can eliminate the coefficient of friction by combining the two equations and then solve for q.

We don’t know the mass of the drops we are using and we don’t have a way of measuring them inside the apparatus.We can calculate the mass, however, if we know the density of the oil and can determine the size of the drops. Tofind the size we will use Stokes’ Law which relates the radius of a spherically shaped object to the velocity at whichit falls in a viscous medium:

a =√

9ηv1

2gρ(8.3)

where a is the radius of the moving object, η is the coefficient of viscosity for the medium in which the object ismoving, v1 is the object’s velocity and ρ is the density of the object. In order to apply Stokes’ Law to our oil dropswe must add a correction factor to the viscosity, η, which gives us an effective viscosity of:

ηeff = η

(1

1 + bpa

)(8.4)

where b is a constant and p is the atmospheric pressure. We now have the necessary equations to calculate the chargeon a drop of oil. First we need to calculate the radius of the drop:

a =

√(b

2p

)2

+9ηv1

2gρ− b

2p(8.5)

We can then find the mass of the oil drop:

m =43πa3ρ (8.6)

8.2. REAL MILLIKAN EXPERIMENT 43

Finally we can find the charge of the drop by combining equations 8.1 and 8.2:

q =mg

E

(1 +

v2

v1

)(8.7)

since we know g, we can calculate m, E is the ratio of the voltage between the plates to the distance between themand we measure v1 and v2. In order to get the charge in coulombs you will need to use SI units (i.e. g in m/s2, ρ inkg/m3, a in m, vx in m/s, η in N·s/m2, p in Pa, etc.) and the constant b = 8.20× 10−3 Pa·m.

You will want to measure velocities for many oil drops so you can perform the same (or a similar) analysis as youdid for the first experiment with the USOs and containers.

Questions:

1. Draw the free body diagrams for the case of an oil drop in the apparatus with and without an electric field andshow that equations 8.1 and 8.2 are correct. Are there cases where these equations are not correct?

2. Fill in the steps that lead to equation 8.5.

3. Why do you think oil is used instead of some other liquid (like water) or a solid?

4. Is your observed value for e within experimental uncertainty of the accepted value of 1.6× 10−19 C?

5. What is your biggest source of uncertainty? Is there a way you can reduce this?

Chapter 9

Physics of Gamma Spectroscopy

Much like atoms, atomic nuclei have a certain number of discrete excited states in which they can exist. Usually,the lifetime of these states is very short (< 1 ns). The excited nucleus falls back to its ground state by emitting aphoton whose energy corresponds to the energy difference between the excited and ground states. Typically, thisenergy difference is in the 100 keV - 3 MeV range.

137 56Ba

03/2+

661.66011/2– 85.1

661.

660

M4

stable

2.552 m

137 55Cs≈

5.6% 12.1

94.4% 9.61

7/2+ 030.07 y

Qβ−=1175.63

Figure 9.1: The decay chain shows that 13755 Cs decays to 137

56 Ba via beta decay, n → p + e− + νe. The 661.66 keVgammas are produced by the subsequent decay of the excited 137

56 Ba to its ground state.

There are two ways to produce nuclei in excited states. In the first case, the nuclei are bombarded with energetic

45

46 CHAPTER 9. PHYSICS OF GAMMA SPECTROSCOPY

60 28Ni

00+

1332.5162+

2158.642+2505.7654+ 2.

0×10

-625

05E4

99.9

011

73.2

37E2

(+M

3)

0.00

7634

6.93

0.00

111

2158

.57

E2

0.00

7682

6.06

M1+

E2

99.9

820

1332

.501

E2

stable

0.713 ps

0.59 ps 0.30 ps

60 27Co≈

0.057% 15.02

<0.0022% >13.399.925% 7.5

5+ 05.2714 y

Qβ−=2823.9

Figure 9.2: 6027Co decays to 60

28Ni via beta decay, n → p + e− + νe, and 99.93% of the time to the 4+ state of 6028Ni.

The 1173 keV gammas are produced by the subsequent decay to the 2+ state, and 99.98% of the time the 1333 keVgammas in the transition from 2+ state to the ground state.

particles, each of which carries an amount of energy that is equal or larger than the energy needed to excite thenucleus. In the second case, the excited states are fed by disintegration of another nucleus. This disintegration mayproceed through β decay, α decay, fission, or some combination of these processes. Figures 9.1 and 9.2 show thedecay processes for two of the commonly used sources, 137Cs and 60Co.

In the case of 137Cs, a neutron in the cesium nucleus turns into a proton via beta decay (n → p + e− + νe), whichchanges the nucleus to 137Ba. However, this barium nucleus is, in 94.4% of the cases, produced in an excited state,in which it has 661 keV more energy than in the ground state. When the barium nucleus falls back to its groundstate, it releases this 661 keV in the form of a photon.

The case of 60Co is somewhat more complicated. This nucleus also decays by β decay, and forms 60Ni in this process.In 99.9% of the cases, the nickel nucleus is produced in an excited state where it has 2505 keV more energy than inthe ground state. However, rather than decaying directly to this ground state, the nuclear disintegration is a two-stepprocess in this case. First, the excited nucleus makes a transition to a state located 1333 keV above the groundstate, then it makes the transition to the ground state. In both transitions, the energy difference is carried away bya γ ray, and therefore a 60Co source emits equal numbers of γ rays with 1173 keV and 1333 keV, respectively. Asindicated in the decay scheme, there are also some other transitions that play a role in this decay but they are quiterare.

In the following experiments, we will concern ourselves with the properties of these radioactive sources and theinteraction of γ rays with matter.

9.1. CALIBRATION AND ENERGY RESOLUTION OF DETECTOR 47

9.1 Calibration and Energy Resolution of Detector

The detectors used to measure the properties of γ rays are based on the principle that some materials emit visible lightwhen their atoms or molecules are excited by ionizing particles, a phenomenon known as fluorescence or scintillation.This scintillation light can be detected and transformed into an electrical signal (pulse). The amount of light (andthus the charge carried by the electric pulse) is in principle proportional to the total energy dissipated by theionizing particles that created it. The scintillation light produced by the detector is transformed into an electricpulse by a photomultiplier tube (PMT). The scintillation photons produced in the NaI detector produce electronsin the photocathode of the PMT, which is optically coupled to the detector, through the photoelectric effect. Thesephotoelectrons are accelerated in the PMT towards a structure of metal plates called dynodes. Typically, thepotential difference that drives this acceleration is 75 V between each set of consecutive dynodes, and since there are8 acceleration stages, the PMT operates at ∼ 600 V. At each dynode, the bombardment with accelerated electronsleads to the emission of more electrons, the number of electrons is thus multiplied at each of the 8 stages. Theresulting shower of electrons at the anode gives rise to an electric pulse, whose amplitude (or total integrated charge)is directly proportional to the number of scintillation photons that started this process. And since the number ofscintillation photons, at least for events in the photopeak, is proportional to the energy of the γ rays emitted by thesource, the amplitude (or the total integrated charge) of the pulses produced by the PMT is proportional to thatenergy.

The objective here is to measure the energies of these gamma rays from various different sources with accuracy. Wefirst need to establish a procedure through which we convert the channel number (counts) to energy (keV). Begin byplacing the 60Co source in front of the detector. Switch the power supply on, and slowly increase the high voltagefor the PMT to its nominal value (4.0 on the 10-turn potentiometer). The generation of charge pulses is indicatedby a flickering red light on the unit.

Record the 60Co spectrum on the computer with the WinDAS program (see manual for details on how to use thisprogram and take time to familiarize yourself with it). You should see the two photopeaks (1173 and 1333 keV)clearly separated. Look at the effect of changing the discriminator level, up and down. When you increase/decreasethe high voltage (i.e. the amplification of the PMT), the two peaks move up/down. Set the high voltage such thatthe 1332 peak is recorded in channel 800. This means that the full range of the ADC (1024 channels or 10-bit range)now corresponds to an energy of ∼ 1700 keV. Set the discriminator level such that pulses are recorded in channel 20and higher.

Now, we are ready to calibrate the signal distribution, i.e. you will have to establish the relationship between channelnumber and γ ray energy. WinDAS has a special program to do this (see manual for details). Although there isno unique way of accomplishing calibration, let’s pick two known peaks 22Na (1275 keV) and 57Co (122 keV) forthis purpose. First record the spectrum of 22Na. When you have accumulated a nice peak at 1275 keV, replace thesource by 57Co and see the 122 keV pop up. When you have accumulated enough statistics, you can proceed withthe calibration, following the program’s instructions. After completion of this procedure, the system is calibrated,i.e. the relationship between channel number and energy is established for all subsequent measurements.

Remove all sources and record a background spectrum, for about 15 minutes to establish the background.

Check the correctness of the calibration with the other available sources. Make sure you save your data files forfurther analyses. Use all of these sources: 22Na, 40K, 54Mn, 57Co, 60Co, 109Cd, 133Ba and 137Cs. Look up the decayschemes for these isotopes in [6] and complete Table 9.1. In completing this table, indicate what kind of a nuclearprocess is responsible for the emission of these γ rays.

Questions:

1. What is the calibration constant for the detector? How many counts equal 1 keV?

2. How well is Emeasi measured? How is the statistical precision of this data point determined?


Table 9.1: Calibration measurements with γ rays. Emeas1 refers to the measured peak position of the most dominant

decay where σ(Emeas1 ) is the measure (standard deviation) of width of the peak. Eacc

1 refers to the accepted valuefor γ energy. The subscript 2 refers to the second most frequent decay mode.

Isotope Decay Emeas1 σ(Emeas

1 ) Eacc1 Emeas

2 σ(Emeas2 ) Eacc

2 Activity(keV) (keV) (keV) (keV) (keV) (keV) (Bq)

22Na40K54Mn57Co60Co109Cd133Ba137Cs

3. Is the detector linear in the measurement range (0 to 1700 keV)? How can you quantify this measure of linearity?How about plotting (Emeas

i − Eacci )/Eacc

i vs Eacci ? How is this plot useful?

4. What is the energy resolution of this detector? In other words, how precisely does this detector measure theenergies of these γ rays? Plot σ(Emeas

i )/Emeasi vs Emeas

i . What would be a good functional form to fit thesedata points?

5. How useful is the background measurement? Discuss.

6. Did you observe a peak appear near the high end of your energy range while you are taking the backgroundmeasurements? What is the energy of these γs and what is the source? What is the rate of these events(count/min)? Are these events statistically significant to be a concern for the above measurements?

7. As indicated in Figures 9.1 and 9.2, in addition to gamma rays from the excited daughter nuclei, an electronis too emitted in the nuclear decay process. Why don’t we detect these electrons?

8. How can we determine the efficiency of the detector?

9. Once the efficiency of the detector is determined, measure the activities of the isotopes in Table 9.1 and compareto what is indicated on the labels.

10. 6027Co has an energetic (2.505 MeV) gamma emission but it is very rare (2.0 × 10−6). Knowing the activityof the source, the efficiency and geometrical acceptance of the detector, for how long should you take data toobserve a statistically significant peak above background? Substantiate your arguments with data.

11. Gamma ray spectroscopy is a useful tool in identifying isotopes in samples. Determine the what the unknown(mystery) isotope is. Discuss the dominant features of the spectrum. What is the activity of this source?

12. There are three rock samples for which the gamma spectra should be obtained and analyzed. Take statisticallysignificant data for hematite, carnotite and the unknown sample. Identify the dominant peaks. Estimate thegamma activity from these rocks. Discuss the natural decay chain. Learn about the chemical composition ofthese rocks. What can you say about the unknown sample?

13. Generate backscattered γs by placing the lead cover on the unit. Use the 137Cs source for this purpose. Recordspectra with and without the cover on (equal times) and subtract the uncovered spectrum from the coveredone. This should reveal the backscattered γs. What is the average energy of these γs? What would you expectthis energy to be?

9.2. ENERGY LOSS BY GAMMA RAYS 49

9.2 Energy Loss by Gamma Rays

The objective here is to investigate the attenuation of gamma rays through materials. First, we wish to understandthe mechanisms through which a γ loses its energy as it travels in a medium and second, we would like to quantifythe material dependence of this energy loss.

In order to clarify the first point, we look at Figure 9.3 which shows the cross sections, in units of barns per atom,for different processes for gamma ray energies from 10 eV to 100 GeV [9]. Note that one barn is 10−28 m2. Thephotoelectric effect (σp.e.) contributes the largest to the total cross section for photons with energies up to 1 MeV.In atomic photoelectric effect, the atomic electron is ejected in the process of absorbing the photon. The Rayleigh(σRayleigh) and Compton (σCompton) scattering are the next. In the case of Rayleight scattering, the scattered photonleaves the atom neither ionized nor excited. In the case of Compton scattering, as we will study in the followingexperiment, the photon scatters off an electron and imparting some of its momentum to the atomic electron. Theproduction of electron-positron pairs requires photon energy at least twice the electron mass (1022 keV) to occur.Thus, pair production in nuclear (κnuc) and electron (κe) fields contribute to the total cross section above 1 MeV. TheGiant Dipole Resonance (σg.d.r.) is when the target nucleus is broken up and not surprisingly this effect is significantwhen the photon energy is in the same order of the nuclear binding energy1.

Photon Energy

1 Mb

1 kb

1 b

10 mb10 eV 1 keV 1 MeV 1 GeV 100 GeV

(b) Lead (Z = 82)- experimental σtot

σp.e.

κe

Cro

ss se

ctio

n (b

arns

/ato

m)

Cro

ss se

ctio

n (b

arns

/ato

m)

10 mb

1 b

1 kb

1 Mb(a) Carbon (Z = 6)

σRayleigh

σg.d.r.

σCompton

σCompton

σRayleigh

κnuc

κnuc

κe

σp.e.

- experimental σtot

Figure 9.3: Photon total cross sections as a function of energy in carbon and lead showing contributions from differentprocesses (see text for details) [9].

In this experiment, we measure the gammas that are removed from the photopeak by the processes described abovewhen a piece of lead is placed between the the gamma source and the detector. We use the 662 keV photons from

1The data for different elements, compounds and mixtures can be found in http://physics.nist.gov/PhysRefData


13755 Cs and the NaI(Tl) detector for this study. The decrease of intensity of gamma rays after they have traveledthrough some material is

I(x) = I0 exp (−µx) (9.1)

where I is the intensity after the absorber, I0 is the intensity before, µ is the total-mass absorption coefficient inunits of cm2/g and x represents the density thickness of the material in units if g/cm2. The density thickness issimply density (g/cm3) multiplied by the thickness (cm) of the absorber.

The experimental procedure is straightforward: once the background and the energy calibration are established,increase the thickness of lead sheets between the source and the detector. Observe for the same fixed measurementperiod how many photon are attenuated. In each step, make sure you have the data files saved.

Questions:

1. Plot intensity (I(x)) vs absorber thickness in units of mg/cm2. Here I(x) should represent the backgroundsubtracted data divided by the live time.

2. From the plot above determine the density thickness of the material that will reduce the original intensity byhalf. More specifically,

ln(1/2) = −µx1/2

x1/2 =0.693µ

(9.2)

What is x1/2?

3. What does this give for µ? The accepted value is 0.105 cm2/g for Pb. Compare these values.

4. Repeat the experiment with aluminum. Determine x1/2 and µ for Al. The accepted value is 0.074 cm2/g forµ. Compare and discuss your results.

5. What is the total cross section at 662 keV for Pb and Al? Express your measurement in barns per atom. Whatis the error in your measurement?

9.3. DEEPER LOOK INTO 60CO SPECTRUM 51

9.3 Deeper Look into 60Co Spectrum

Let’s understand the fundamental features of the 60Co isotope. The decay chain is shown in Figure 9.2 and a samplespectrum is given in Figure 9.4.

0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

3500

4000

Energy (keV)

Eve

nt/2

.85

keV

0 500 1000 1500 2000 2500 3000100

101

102

103

104

Energy (keV)E

vent

/2.8

5 ke

V

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

Energy (Count)

Ene

rgy

(keV

)

0 500 1000 1500 2000 2500 3000−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Energy (keV)

Per

cent

age

Diff

eren

ce F

rom

Str

aigh

t Lin

e

Figure 9.4: Top left plot is the measured spectrum of 60Co isotope with a NaI(Tl) detector in linear scale in ordinate.Two photopeaks are clearly visible at 1173 and 1333 keV. Top right plot is the same spectrum but plotted inlogarithmic scale where a clear third sum-peak is visible. The bottom left curve shows the fitted calibration curve ofthe form y = mx + b between the measured counts from the detector to energy (keV) units using the three knownpeaks. The bottom right plot displays the precision of this calibration curve where the percentage difference betweenthe fitted curve and the data points are shown.

The calibration curve in principle should cover the widest possible energy region from a known lowest energy to thehighest. The curve with three points in Figure 9.4 is for illustration purposes only.

In addition to these three photopeaks, there are distinct features at lower energies. There is a clear shoulder below1000 keV and although not clear from the spectrum there is another shoulder below the 1173 keV photopeak.These are where an incoming photon from 60Co undergoes a head-on Compton scattering at θ = 180◦ and transfers


maximum energy to an atomic electron in the detector. Since there are two photons with distinct energies from thesource, we get two what is called Compton edges and these correspond to the maximum energy that an electron canhave from this scattering. The events lower than these edges are the photon-electron interactions at all other angles.There are many events lower than 100 keV (X-ray peaks) which are generated by photoelectric absorption withinthe material surrounding the detector.

Questions:

1. Take several spectra from different sources

2. Plot the calibration curve with data points from all the identifiable photopeaks (see Figure 9.4 bottom left).

3. Calculate and plot percentage difference from straight line (see Figure 9.4 bottom right).

4. Derive the maximum possible energy transfer to an atomic electron from an incoming photon using conservationof energy and momentum. Check if these correspond to the shoulders or Compton edges.

5. Calculate the Compton edge for gammas at 376 keV (133Ba), 661 keV (137Cs) and 1258 keV (22Na) and comparethem with your measurements.

6. Again from the conservation laws, derive the expression for the energy of the scattered photon from an electronas a function of the energy of the incoming photon and the scattering angle, θ. This is the Compton scattering.

7. The 60Co photons are essentially isotropic. So, some photons will backscatter into the detector. At what energyshould we expect these back scattered photons?

8. If the energy of a photon above twice the mass of electron (1022 keV), there is a chance that electron-positronpairs are created. If this is the case (as is in 60Co), we expect to observe two additional peaks in the spectrumwhere the positron annihilates with an electron and decays back to two 511 keV photons. If we capture bothof them in the detector, we should observe a peak at 1022 keV, if we catch only one, it should appear at 511keV. Analyze your data and discuss if this makes sense.

Chapter 10

Physics with Cosmic Muons


Cosmic radiation is high energy electromagnetic waves and energetic charged and neutral particles that are created incollisions of highly energetic galactic particles (mostly protons) with atomic nuclei in the Earth’s upper atmosphere.In these collisions, large numbers of secondary particles (air showers) are generated. These are mostly charged andneutral pions, and they are short-lived. When at rest, π± decays into a µ± and a νµ in about 2.6× 10−8 seconds, or26 ns. Neutral pions decay into two photons even faster, in ∼ 10−16 seconds. When pions are traveling at relativisticspeeds, because of time dilation as postulated by Einstein and later measured with high precision, they live longerby a factor γ:

γ =1√

1− v2

c2

. (10.1)

where c is the speed of light and v is the pion’s speed. In terms of their energy and mass, we can recast Equation10.1:

γ =E

mπc2=K +mπc

2

mπc2. (10.2)

where K is the kinetic energy of the pion, and mπ is its mass. Thus, the possible values for γ range from 1 (pion atrest) to perhaps several thousands for highly energetic pions in air showers. We can simply estimate the distance dtraveled by a pion in SI units for γ = 103.

d = γvt ≈ γct = (103)× (3× 108 ms

)× (26× 10−9s) ≈ 8000 meters (10.3)

Since proton-nucleus interactions occur 20 to 40 km in altitude, pions in general do not reach the Earth’s surfacebut instead decay into muons (µ) and neutrinos (ν).

While neutrinos undergo weak interactions, muons interact electromagnetically. Neutrinos go through the Earthessentially unnoticed, but the muons start losing their energy through ionization before they decay into a pair ofneutrinos and an electron (µ− → νµ + νe + e−). Muons, when at rest, live about a hundred times longer comparedto pions: 2.2 × 10−6 seconds, or 2.2 µs. Thus, if they are energetic enough they will travel many kilometers beforethey decay. That’s why we observe many muons at sea-level.

Questions:

53

54 CHAPTER 10. PHYSICS WITH COSMIC MUONS

1. What are the origins of cosmic radiation? Where do they come from?

2. What is Birk’s law?

10.2 Detection of Cosmic Muons

There are many ways of detecting cosmic muons that have been developed over the course of the last century.Early in the 20th century emulsion plates (much like photographic film) were developed. These films, sensitive tocharged particles, were used to study cosmic radiation and many of the elementary particles that we know today werediscovered using this detection device. One of the major drawbacks of this approach is that the stack of emulsionplates are exposed for a long time, thus they integrate many events over time. If one wants to study the cosmic rayevents quickly (or one at a time) we resort to scintillation counters as in these experiments.

Scintillation counters have been developed for very fast detection of cosmic rays. Scintillation material is a transparentmedium such as plastic that is doped with fluorescing dye molecules. These fluorescent molecules get excited bythe radiation and subsequently decay (de-excite) by emitting photons in the UV and/or visible region. The photonsbounce around inside the scintillator and may exit in a certain direction to be detected by photomultiplier tubes(PMTs). As the name implies, PMTs are photo-sensitive devices based on the photoelectric effect (Einstein again!).They output an electrical pulse that is fast (a few tens of ns) and is proportional to the number of photons incidenton its photocathode. An electron in the photocathode material may absorb the incoming photon from the scintillatorand gain enough energy to leave the photocathode as a photoelectron. The efficiency to liberate such photoelectronsvaries with PMTs but is typically 10 to 20 %. Once the photoelectron is freed it is accelerated toward multiplicationstages (dynodes). Each electron that hits a dynode can liberate several secondary electrons from it’s surface. APMT with 12 such multiplication stages has a gain factor G of ∼ 312, or ∼ 105. Thus a single photon seen by a PMTmight generate a total charge of

Q = Gqe = (105)× (1.6× 10−19)C = 16 fC (10.4)

When observed on an oscilloscope this charge corresponds to about a millivolt. Note that this is the voltage pulsefor a single photon. Typically thousands of fluorescent photons are created when a charged particle traverses thescintillator (a 1 MeV electron will produce roughly ten thousand photons in a scintillator like ours) which ultimatelyleads to hundreds of photoelectrons, i.e. several hundred millivolts on the oscilloscope.

Here are a few things to keep in mind when dealing with the PMTs.

1. The PMTs are extremely sensitive to light. Never expose the PMTs to ambient light and never pull thesockets (bases) off the PMTs when they are powered.

2. The Fluke high voltage supply biases all four PMTs through a passive splitter with the same negative voltage.Do not change the polarity of the supply.

3. Be careful and do not get shocked by the HV. The power supply can provide a large current.

4. Do not exceed −1500 V. You can easily damage the PMTs.

5. You first turn on the power switch (left) and then the standby/on switch. HV will not be delivered to thePMTs until the standby light is on, and you flip the second switch to the on position.

10.3. COSMIC RAYS IN LUBBOCK 55

10.3 Cosmic Rays in Lubbock

In Section 10.1, a brief discussion of cosmic rays is presented. For more details see [9, 14, 20, 22]. Figure 10.1 showsthe expected vertical rate of cosmic particles as a function of altitude. The approximate rate of muons is roughly 70m−2 s−1 sr−1 at sea-level. The neutrino rate is about twice that of the muons. You may findhttp://www2.slac.stanford.edu/vvc/cosmicrays/crslac.html useful. You can also download CORSICA fromhttp://www-ik.fzk.de/corsika/ which is a common simulation tool for air showers among astrophysicists.

Figure 10.1: Vertical fluxes of cosmic rays in the atmosphere with E > 1 GeV estimated from the nucleon flux. Thepoints show measurements of negative muons with Eµ > 1 GeV (from [9]).

Questions:

1. When a muon with Eµ ≈ 1 GeV passes through the scintillator block, how much energy does it lose?

2. How many photons are generated in the scintillator?

3. How much charge is detected per muon by the PMTs? Do these numbers make sense? Give quantitativeinformation.

4. What is the discriminator threshold for each channel? Test the effects of different levels (be careful with thetrim pots) of thresholds. These levels strongly affect your measurements, explain why.

5. What is the (average) cosmic ray rate (in Lubbock)?


6. Are your results statistically significant? How do you know?

7. Are your results consistent with what others have measured? Where does your data point fall in Figure 10.1?

8. Is there a day/night difference in cosmic muon rate? If so, how much? Is this significant?

9. Do you observe any significant deviation in the muon rate over a period of a week or more?

10. What type of particles are you actually measuring? How do you know?

11. Can you measure the flux at a different altitude where you might have a statistically significant (and a different)result? Rooftop? Carlsbad caverns?

12. Measure the cosmic ray flux as a function of angle with respect to the vertical. Explain in detail your experi-mental setup, method and results. Be sure to discuss the statistical and systematic errors (see Figure 10.2 asan example).

13. Analyze your flux vs angle data by fitting it with a function. What are your criteria in choosing a fit function?What does it mean?

14. Is there an effect from electronic noise, radioactivity from the walls or something else that may mimic a cosmicray in your experiment? How do you quantify these background events? What is the accidental event rate?

Figure 10.2: The angular distribution of the cosmic ray rate is symmetric around the vertical direction. The solidline is a fit of the form cos2 θ.

10.4. ENERGY SPECTRUM OF COSMIC MUONS 57

10.4 Energy Spectrum of Cosmic Muons

In this experiment, we will measure the energy spectrum of vertical cosmic muons. Once the cosmic ray countersare setup in coincidence mode with about a meter between them, the mass in between the counters is incrementallyincreased and the cosmic ray rate is measured. In other words, we would like to see how much material it takes tostop the cosmic muons. Figure 10.3 schematically shows the setup.

Muon

Lead Bricks

Top Counter

Bottom CounterSupport

Structure

Floor

~1 m

Figure 10.3: The setup to measure the spectrum of vertical cosmic muons.

The amount of energy lost by a muon going through lead is about 12.4 MeV/cm. Therefore, by using lead bricksone meter of lead will stop 1.24 GeV muons on average.

Questions:

1. Plot the vertical cosmic muon rate vs absorber thickness. Make sure the lead bricks completely cover the areaof the detectors and you measure the event rate one lead brick layer at a time. What does this tell you aboutthe energy spectrum of the muons?

2. Do you think there is a difference in the energy spectrum of vertical and horizontal cosmic muons? Why orwhy not?

Bibliography

[1] C. Amsler. The determination of the muon magnetic moment from cosmic rays. American Journal of Physics,42:1067, 1974.

[2] H. Bethe. Theory of the passage of fast corpuscular rays through matter. Annalen Phys., 5:325–400, 1930.

[3] J. S. Blakemore. Solid State Physics. Cambridge University Press, 1985.

[4] John M. Blatt and Victor F. Weisskopf. Theoretical Nuclear Physics. Dover Publications, October 1991.

[5] A. K. Das and T. Ferbel. Introduction to nuclear and particle physics. River Edge, USA: World Scientific (2003)399 p.

[6] Richard B. Firestone. Table of Isotopes. Wiley-Interscience Publication, John Wiley and Sons, Inc., 8 edition,1996.

[7] L. Gonzales-Mestres. Properties of a possible class of particles able to travel faster than light. In 30th Rencontrede Moriond: Dark Matter in Cosmology, Clocks and Tests of Fundamental Laws, pages 645–650, 1996.

[8] D. Green. The Physics of Particle Detectors. Cambridge University Press, 2000.

[9] Particle Data Group. Journal of Physics G, 33, 2006.

[10] Glenn F. Knoll. Radiation Detection and Measurement. John Wiley and Sons, 1989.

[11] S. Kenneth Krane. Introductory Nuclear Physics. John Wiley and Sons, Inc., 1988.

[12] William R. Leo. Techniques for Nuclear and Particle Physics Experiments. Springer-Verlag, 1987.

[13] M. Ali Omar. Elementary Solid State Physics. Addison Wesley, Inc., 1978.

[14] Donald H. Perkins. Introduction to High Energy Physics. Addison-Wesley Publishing Co., 1985.

[15] Philips Photomultipliers. Photomultiplier Tubes – Principles and Applications. Philips Photonics, 1994.

[16] Robert F. Pierret. Advanced Semiconductor Fundamentals. Prentice Hall, 2nd edition edition, 2002.

[17] M .A. Preston and R. K. Bhaduri. Structure of the Nucleus. Addison-Wesley Publishing Company, 2 edition,1975.

[18] E. Recami. Classical tachyons and possible applications: A review. Riv. Nuovo Cim., 9:1–178, 1996.

[19] S. Rosenblum. J. Phys. Radium, 1:1, 1930.

[20] Bruno Rossi. High-energy Particles. Prentice-Hall, 1952.

[21] E. Rutherford and H. Geiger. Proc. Roy. Soc (London), A81:141, 1908.

59

60 BIBLIOGRAPHY

[22] Emilio Segre. Nuclei and Particles. W. A. Benjamin, Inc, 1965.

[23] R. M. Sternheimer. Phys. Rev., 88:851, 1952.

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