Download A Guide for the Perplexed Experiments in Physics (Version 4.0

A Guide for the Perplexed
Experiments in Physics
(Version 4.0 Spring 2009)
Nural Akchurin, Mohammad Alwarawrah, Ken Carrel, Ross Carroll
Texas Tech University,
Department of Physics,
Lubbock, TX, USA
January 24, 2009
2
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
3.1
Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2
Experimental Procedure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 The Franck-Hertz Experiment
4.1
21
Background Information and Performing the Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Radioactivity
25
5.1
Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2
Experimental Procedure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
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
6.1
Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2
Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.3
6.2.1
Basic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2.2
Measurements (general) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.2.3
Helium standard spectrum
6.2.4
The Balmer lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
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
7.1
Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.2
Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
8 Millikan Experiment
39
8.1
Experimental Procedure
8.2
Real Millikan Experiment
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
9.2
Energy Loss by Gamma Rays
9.3
Deeper Look into
60
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Co Spectrum
10 Physics with Cosmic Muons
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
53
10.1 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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.
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 Wikipedia
webpage). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
9.1
137
−
The decay chain shows that 137
55 Cs decays to 56 Ba via beta decay, n → p + e + ν̄e . The 661.66 keV
137
gammas are produced by the subsequent decay of the excited 56 Ba to its ground state. . . . . . . . . 45
9.2
60
27 Co
9.3
Photon total cross sections as a function of energy in carbon and lead showing contributions from
different processes (see text for details) [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
9.4
Top left plot is the measured spectrum of 60 Co 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 in logarithmic scale where a clear third sum-peak is visible. The bottom left
curve shows the fitted calibration curve of the form y = mx + b between the measured counts from
the detector to energy (keV) units using the three known peaks. The bottom right plot displays the
precision of this calibration curve where the percentage difference between the fitted curve and the
data points are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
. . . . . 34
−
+
60
decays to 60
28 Ni via beta decay, n → p + e + ν̄e , and 99.93% of the time to the 4 state of 28 Ni.
+
The 1173 keV gammas are produced by the subsequent decay to the 2 state, and 99.98% of the time
the 1333 keV gammas in the transition from 2+ state to the ground state. . . . . . . . . . . . . . . . . 46
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 solid
line 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. E1meas refers to the measured peak position of the most
dominant decay where σ(E1meas ) is the measure (standard deviation) of width of the peak. E1acc refers
to 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 development of these experiments. Ken Carrell has been carefully working through these experiments before we consider
them for the students. Mohammand Alwarawrah did the same in Fall 2008. Many TTU undergraduates worked
on 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 programs
are 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 been
measuring (or attempting to do so) for a few hundred years. In this lab you will be following in the footsteps of
many great scientists by making your own measurement of how fast light travels. In addition, you will measure the
speed 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 measure
c using lamps and shutters operated by lab assistants at some fixed distance apart, measured mostly the reaction
time 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, orbits
the 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 changing
pattern of Io’s eclipses to determine how fast light travels. Using this technique a determination of the speed of light
was 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 setting
by using complicated setups. At the current time we have electronics and other tools available to us that can be
used to greatly simplify this measurement and determine c with a decent precision. Using lasers or some other fast
light source and fast photodiodes it is possible to generate and measure pulses of light with durations on the order
of nanoseconds. This means that if we can make light travel some known distance and measure the duration of the
trip, 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 photodiode
can measure.
2.2.
2.2
EXPERIMENTAL SETUP AND PROCEDURE FOR C
13
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), a
beam splitter and a photodiode are all enclosed in a single box. A lens and reflector must be used to direct light
along a certain path and an oscilloscope will be used to measure the results. Before we can perform the experiment
we 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 in
two 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 beam
by the lens L. This light is reflected back by reflector T1 that has been placed at some known large distance s from
the light source. This reflected beam travels backwards on the same path it came from and gets reflected into D by
S. Since the path length of the two legs of the split beam are not the same we will see a time difference in the pulses
measured 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 will
be reflected back into the photodiode. Use the “trigger” output from the LED/detector box as an external trigger
to the oscilloscope. Once pulses are being seen on the oscilloscope, adjust the reflectors so that the amplitudes of
the two pulses are roughly equal. Now a time difference can be measured between the two peaks. An alternative to
14
CHAPTER 2. THE SPEED OF LIGHT AND SOUND
having both pulses shown at the same time is to block light from first one leg and then the other and measuring the
position 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 different
distances s.
Questions:
1. Which method of measuring the time difference (using the two peaks or blocking one leg and measuring one
at 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 all
your 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 value
correct within errors?
4. How would your measurements and results differ if some transparent material (like glass) was placed in the
light path between the LED/detector box and T1 ?
2.3.
2.3
SPEED OF SOUND IN SOLIDS
15
Speed of Sound in Solids
The purpose of the experiments here is to measure the speed of sound in solids accurately and understand the reasons
why the speed of sound is higher or lower in a given material. The experimental setup consists of a 1-m long metal
rod which is connected to piezoelectric crystals at both ends. When an impulse is generated at one end, the time
difference between the two signals gives the time of travel for that sound wave. The piezoelectric crystals for this
setup 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 experimental
precision 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 make
your measurement.
6. Develop a classical model that describes how the sound travels in a medium that is consistent with your
measurements 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 we
measure. Find out how many volts would be generated per meter for these crystals? Explain your numbers.
16
CHAPTER 2. THE SPEED OF LIGHT AND SOUND
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 and
compare it to c. Using a long coax cable (RG58U), a pulse generator and an oscilloscope, we can determine v. The
method of measurement is much the same as the previous measurements of speed of light and sound in media, but
in 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 characteristic
impedances?
Chapter 3
The Photoelectric Effect and Planck’s
Constant
When light is incident on a metal surface, electrons bound within the metal can be released in a process known
as the photoelectric effect. This process does not follow classical physics predictions and helped lead the way to a
better understanding of the quantum world. In this lab you will use the photoelectric effect to help determine the
fundamental constant related to quantum physics.
17
18
CHAPTER 3. THE PHOTOELECTRIC EFFECT AND PLANCK’S CONSTANT
3.1
Background Information
The photoelectric effect has both historical and practical significance. Experimental observations of the photoelectric
effect were one of the things that led Einstein to come up with the idea of the photon and this effect is the basic
process 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 work
function, φ, of the material. This energy is typically on the order of a few electronvolts. If one analyzes this
process from a classical physics perspective one comes to the conlusion that there is a certain minimum intensity an
electromagnetic wave would need to transfer enough energy to release electrons from a metal. One would also expect
that the energy of an electron released from the metal by this process would depend on the intensity of the light wave
and not the frequency of the light. The experimental results, however, proved this line of thinking to be false and
instead 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 to
the intensity of the light. And lastly, the energy of the emitted photoelectrons increases with the frequency of the
incident light.
To adequately describe what was happening, Einstein made the assumption that light interacts with the electrons
found in the metal just as a stream of particles would. These “particles” of light he named photons and possessed a
well-defined energy given by E = hf where f is the frequency of the light and h is a fundamental constant known as
Planck’s constant. In this case the maximum amount of kinetic energy a photoelectron could obtain would be the
difference between the energy of the light and the work function of the material:
Emax =
h
f −φ
e
(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 the
energy 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 frequency
range?
3.2.
3.2
EXPERIMENTAL PROCEDURE
19
Experimental Procedure
For this experiment you will be using a mercury lamp, a monochromator and a photocathode inside a box with
an 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 the
monochromator. Open the slits of the monochromator so that you can easily see light passing through and change
the wavelength of light allowed to pass through the monochromator until you find the three very bright lines. Record
these 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 few
minutes. After this, use the appropriate knob to adjust the ammeter so that it reads exactly zero. Using the provided
stands align the three pieces of equipment so that light passes from the lamp through the monochromator and onto
the photocathode. You will be recording values for the photocurrent using the ammeter on the box enclosing the
photocathode. Values for the opposing voltage can be found by using the connectors on the photocathode enclosure
and a multimeter.
You will perform the experiment in two ways and judge which method is the best/easiest to perform. The first
method will be to adjust the slits on the monochromator so that each of the three spectral lines have the same initial
photocurrent (i.e. when there is no opposing voltage). From there, adjust the voltage so that the ammeter is on a
particular current and record the current and voltage. Make measurements for several photocurrent values all the
way down to zero, which is known as the stopping potential. The second method will be similar, but instead of
adjusting the slit for each spectral line you will set the slit width using the blue line and use this same width for the
other two spectral lines.
Taking the stopping potential for each spectral line as Emax in equation 3.1 and the wavelengths you found for the
spectral lines of the mercury lamp, graph your results and determine both the ratio h/e and φ. You can expect
something 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, name
each of the 7 key components to this experiment labelled ’A’ through ’G’ in figure 3.2. Hint: How is a current
produced 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 from
the 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 experimental
value.
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, explain
how.
8. Could this experiment be performed with a light source that did not have discrete emission lines? Why or why
not?
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 were
awarded the Nobel Prize in Physics for it in 1925. At the turn of the 20th century, physics of the small scale was
still a very new and poorly understood field. There were two competing theories on atomic structure: Rutherford’s
Theory and Bohr’s Theory. The theories were similar but differed in a few key aspects. This experiment provided
clear evidence in support of one of these two theories.
21
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) and
the current (I):
V = IR
If we replace the usual current carrying device - a copper wire - with something else do we expect the same relation
to 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 will
notice three things inside it. There is a cathode, a grid anode and a collector electrode. The grid anode is kept
at a positive voltage with respect to the cathode so that electrons are accelerated toward the grid. The collector
electrode is kept at a slightly negative voltage so that only electrons with a certain energy threshold can reach it. On
the front of the heating oven for the mercury tube is a wiring diagram and connectors that are labelled so wiring to
the power supply can be done easily. When connected to the tubes, the power supply will provide a current through
the tubes by accelerating electrons through a certain voltage. It measures the current by determining how many
electrons make it through the gas to the collector electrode. This allows us to determine how voltage and current
are 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 you
have it wired correctly and ask if you have questions - this equipment can be damaged if powered on with a wrong
connection) and allow the tube to warm up before taking any measurements. Set the switch in the middle of the
power 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 the
various 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 many
maxima 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) or
2) watch the oscilloscope to find the local maximum or minimum. As you adjust the voltage on the power supply
watch the tube through the window. There should be small areas of light visible: note their position, intensity and
size as you adjust the voltage. After determining the voltages of the maxima and minima of the current, use the
provided 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 heated
so that the vapor inside can be at the right pressure for the phenomenon we are after to be seen. Adjust the
temperature (between ∼150 and ∼200 degrees) as well as the knobs on the power supply to obtain the best curve
on the oscilloscope.
IMPORTANT: Both tubes are fragile and can be damaged easily. Also, the mercury tube gets very hot so do not
allow 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 oscilloscope
look 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 is
different?
3. Briefly explain the theories proposed by Rutherford and Bohr to describe atomic structure. In what ways are
the two theories different?
4. Describe the graph you obtained and try to explain its features using physical arguments (it may be useful to
include how the adjustments made with the knobs affected the graph and how the features of the visible light
emitted 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 mercury
and 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 discoveries
were made in the physics of the very small scale. One of the discoveries that has had possibly the greatest impact
was radioactivity. Numerous wonderful (and a few regrettable) applications have been found for these remarkable
discoveries.
25
26
5.1
CHAPTER 5. RADIOACTIVITY
Background Information
The credit for discovering radioactivity is usually attributed to Henri Becquerel who found by accident that uranium
exposes photographic plates. Immediately following this discovery, work by Marie and Pierre Curie as well as Ernest
Rutherford showed that radioactivity was much more complicated than had first been thought and that new physics
would 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 numerous
different configurations inside the nucleus. Elements are determined by the number of protons found inside the
nucleus, but the number of neutrons can vary. These atoms of the same element that differ in the number of
neutrons are called isotopes. There are several forces at work inside a nucleus so it comes as no surprise that some
configurations of protons and neutrons are not stable. When this instability occurs, the system naturally falls into
a stable configuration and this is accomplished through radioactive decay. A single element may have one or many
stable 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 as
radiation. The radiation is either some particle (or particles) or is an electromagnetic wave. At the atomic level this
process 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, we
expect that the number of decay events dN in some time dt should be proportional to the number of isotopes present.
So,
−dN = N λdt
where λ is some proportionality constant. The above is simply a first-order differential equation whose solution is
known 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 case
is the decay constant of the exponential decay and is usually replaced by either the mean lifetime, τ :
τ =
1
λ
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 2
= τ ln 2
λ
(5.2)
The half-life of a radioactive isotope has very important consequences and can be used to distinguish different
radioactive 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 different
radioactive 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 reads
radioactivity rates in disintegrations per minute (or counts per minute - CPM) or something similar. Other typical
units 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
5.2
Experimental Procedure
5.2.1
Measuring the Background
27
Anywhere you are on the Earth there is always some amount of radiation that passes through a given area in a
given amount of time. This is known as background radiation. Typical sources of this radiation include cosmic rays
and naturally radioactive materials found in building materials. You first want to find how much this background
is in the area you are going to perform the rest of your experiment. Turn on the Geiger counter and count the rate
for some short amount of time (you will probably have to count the actual audible clicks of the counter instead of
reading the meter). Using the result from your short measurement, estimate how long you will have to count in order
to get a rate that is accurate to ∼5% and then count for that period of time to get your final background value. This
value (with it’s uncertainty) will need to be subtracted from all of your other measurements as a known background
to your measured signal.
Questions:
1. Determine the background radiation rate for several different probe orientations and at different locations
within the room. Are any of your results significantly different? If they are different, suggest a reason why this
is 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 either
take the measurement from the meter or count the audible clicks as you did for the background measurement. Be
sure 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 in
different 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 this
question 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, determine
the half-life for each of the radioactive sources using equations 5.1 and 5.2. How close are your calculated values
to the ones listed on each of the sources? (Be careful to estimate your uncertainties and think of any systematic
uncertainties 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 the
radioactive 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 an
appreciable 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. Choose
one 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 in
measured activity for different orientations of the source be sure to carefully align the source at each distance (this
is 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 emitted
from the source must pass through this material to be detected. This allows for some of the particles to interact
within the material and possibly not make it out the other side to be detected. The interaction in different materials
is very important when we want to consider how best to isolate a radioactive source so that the radiation does not
cause harm to anyone. For each of the measurements you make, set one of the more highly active sources a fixed
distance from the probe and then place different materials between the source and the probe without changing the
distance between the two. Using several different materials (aluminum, lead, paper, etc.) take measurements using
different thicknesses of the chosen material.
Questions:
1. Make a plot of the normalized intensity (the intensity with nothing between the source and probe divided
by the intensity with something between the source and probe) versus thickness of material for each of the
materials 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 a
reason 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 there
is 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
6.1
Background Information
The constant R in the formula
1
1
1
= R ( 2 − 2)
λ
n2
n1
(n1 > n2 )
(6.1)
which expresses the wavelengths of the electronic transitions in the hydrogen atom is one of the most accurately
determined quanties in science. Its value is known to better than 10−9 , one part in a billion. In this lab, you will
measure 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) and
determined R in that way, it was not until the quantum theory of Niels Bohr (1913) that this constant could be
calculated in terms of other measured quantities. Physically, R is related to the energy required to ionize a hydrogen
atom (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 the
wavelengths of three of the Balmer lines of hydrogen. These measurements will be made more accurate using a
technique that is quite common in spectroscopy: comparison of the unknown wavelengths (hydrogen in this case) to
the well-known wavelengths of a standard spectrum (helium in this case). In modern laser spectroscopy, the standard
spectral lines are often provided by uranium, molecular iodine or molecular tellurium, all of whose visible lines have
wavelengths 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 determine
R.
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 the
diffraction 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 the
angle for different diffraction orders and use equation 6.2 to determine d (since the wavelength is known for
the 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 through
the slit. Adjust the lamp position by staring directly into the collimating scope lens and moving the lamp so
as to get the most light through.
4. To focus the collimating telescope, point the viewing scope directly into the collimating scope, so that you see
an image of the slit. Adjust the focus of the collimating scope until the image of the slit is sharp. Rotate the
slit 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). You
will 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 of
the 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 the
grating.
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 move
the turntable during the rest of the experiment.
6.2.2
Measurements (general)
The grating equation is
d 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 diffraction
angle (also measured with respect to the normal of the grating), and λ is the wavelength of the diffracted light. For
the value of d of our grating and the geometry in which this grating is used, n equals 1 for all the lines you will
observe 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 taking
any measurements and do not move it after. When recording the angles of diffraction be sure to take into account
the 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
Purple
Blue
Green 1
Green 2
Orange
Wavelength (nm)
447.15
471.31
492.19
501.57
587.56
Measure the angle (θλ ) of the diffracted light of each of these lines as accurately as you can. Make good use of the
vertical crosshair. It may be necessary to slightly refocus the viewing oscilloscope to see the images of the slit as
clearly 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 center
the 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. This
procedure ensures that the hydrogen lamp is at the same location that the helium lamp was during the measurements
of the helium reference lines.
Now carefully measure the angles (θλ ) for each of the three visible Balmer lines: The red Balmer-α line, the bluegreen 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, and
in the second column the corresponding measured values for sin (θλ ). Fit the wavelength column to the following
formula, 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), in
which θd = θλ . The additional, quadratic term is needed to compensate for imperfect placement of the grating and
other 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 the
measured 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 your
measured data in columns. The first column contains your measured values of sin (θλ ) for the Balmer lines. Using
your calculated values of a, b and c, you can now calculate the wavelengths of these lines with formula 6.3. Recall
the Balmer formula
1
1
1
(6.4)
= R ( 2 − 2)
λ
2
n
where n = 3 corresponds to Balmer-α, n = 4 to Balmer-β and n = 5 to Balmer-γ. Enter the values of the appropriate
integers, 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. the
inverse of the values in column 1). Fit the columns containing 1/n2 and 1/λ to
y = a0 + b0 x
(6.5)
where y stands for 1/λ and x for 1/n2 . Record the values of a0 and b0 and their standard deviations. Also record the
calculated 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 a0 corresponds to R/4 and that b0 = −R.
Questions:
1. Calculate R from a0 and b0 , along with their associated uncertainties (from the standard deviations). Do
your two values agree with each other, and with the accepted value (R = 1.0968 · 107 m−1 ), to within the
experimental 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 method
of 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 grooves
for 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. Which
of your calculated wavelengths differs most from the actual value? Why do you think that is?
Chapter 7
X-ray Scattering
(Bragg reflection)
7.1
Background Information
In 1895, X-rays were discovered by Wilhelm Röntgen. 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-ray
production 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 orbits
of the atoms. However, a sufficiently violent disturbance can also excite or even remove electrons from the inner
shells. If such an electron is removed, the atom is left with a vacancy in a low-lying excited state. In that case, an
electron from a higher orbital may “jump” to this vacant position and emit a photon in the process. The emitted
photon carries an energy equal to the difference between the energy of the atom in its initial state (with a vacancy in
the 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 an
X-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 atoms
will be removed by bombarding this target with a stream of highly energetic electrons generated in a strong electric
field. Once an inner electron has been removed, a variety of transitions may occur and cause the atom to relax back
to 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 target
material. 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 spaced
sublevels. This phenomenon, called fine structure, is a result of magnetic interactions. However, the energy differences
between those sublevels are too small to be resolved with the apparatus for this experiment and the effects leading
to 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 than
for 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 a
crystal lattice with a regular atomic pattern can be utilized as a type of diffraction grating for X-rays. The method
we will use was proposed by W.L. Bragg in 1912: X-rays incident at a certain angle θ are scattered by parallel planes
of 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 constructive
interference, the path length difference between radiation reflected from successive layers must be an integral multiple
of 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 facecentered 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
s
M
1
3
or
d= 3
d =
(7.2)
2ρNA /M
2ρNA
Bragg reflection is an extremely useful phenomenon in the field of X-ray spectroscopy. It allows separation of X-rays
with different wavelengths much like a diffraction grating inside a monochromator separates the various colors of
visible light. The beam of X-rays generated in our apparatus contains a whole variety of different wavelengths (see the
spectrum in fig. 7.1). However, when this beam is incident at an angle θ to the crystal planes, only the wavelengths
that 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, and
so on. By varying the angle of incidence and measuring the intensity of the Bragg reflection, we can determine the
spectrum of the X-rays generated in the copper target. The purpose of this lab is to measure the wavelengths of the
copper 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) should
be 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 range
from 10◦ to 120◦ will turn out to be sufficient. An example plot of data with and without a filter can be found in
figure 7.3.
7.2
Procedures
• Mount the NaCl crystal in the crystal post ensuring that the major face having “flat matte” appearance is in
the 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.
36CHAPTER 7. X-RAY SCATTERING
(BRAGG REFLECTION)
Figure 7.3: Experimental data showing both the first and second order Bragg peaks for cases with and without a filter.
• Mount the Geiger–Müller 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 the
intensity 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 and
35, 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 which
peaks differ only in amplitude and not in angle. For these peaks, determine θ. Let the order of reflection be
one (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 from
the n = 2 case, then this tells you that you have observed a first and second order Bragg reflection for this
particular 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 to
these lines? Are the observed wavelengths (energies) within the experimental uncertainties equal to the known
values?
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 what
the unknown crystal is. Are the observed spacings within the experimental uncertainties equal to the known
spacings?
8. Does the orientation of any of the crystals matter in this experiment? Explain why or why not.
38CHAPTER 7. X-RAY SCATTERING
(BRAGG REFLECTION)
Chapter 8
Millikan Experiment
Robert Millikan experimentally determined that electric charge is quantized or it came in lumps of a certain minimum
size. 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 of
the droplets, he found the total charge on the droplets. After measuring many droplets, he determined that the total
charge on any droplet was always some multiple of 1.6 × 10−19 C. This is the charge on a single electron and as far
as we know, it is the smallest unit of charge available under normal circumstances.
39
40
8.1
CHAPTER 8. MILLIKAN EXPERIMENT
Experimental Procedure
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 unknown
small object (let’s call it USO). There are unknown number of USOs inside sealed containers. Just as Millikan had
no way of knowing ahead of time the number of electrons on an oil drop, so you will not know the number of USOs in
a given container. By measuring the mass of enough containers, you can determine the most likely mass of a single
USO. 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 which
should end up giving you mUSO ± ∆mUSO . Write a short summary report (no more than 2 pages) which includes
an analysis of the uncertainty of your results. Obviously, you may consult references in error analyses.
8.2.
REAL MILLIKAN EXPERIMENT
8.2
Real Millikan Experiment
8.2.1
Setup and Technique
41
The real experiment that was performed by Millikan (and that you will be doing) is exactly analogous to what was
done in the previous section. In this case, though, the containers will be replaced with drops of oil and the USOs
will be replaced with electrons. The measurements will be done in a different way but the analysis will be identical
to 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 be
used to inject small drops of oil between two parallel plates that can have a voltage applied between them. A light
source illuminates the area between the plates and a telescope is positioned so that one can see the drops moving
between the plates. (NOTE: The image when looking through the telescope is inverted so if a drop is moving up in
the eyepiece it is actually moving down.) The first thing that must be done is to figure out the spacing between the
parallel 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 Wikipedia
webpage).
Now take a little while to get used to spraying drops into the apparatus and observing their motion with the electric
field on and off. Selecting a single drop from a large group can be accomplished by varying the field so that all the
other drops reach one of the plates. There are two measurements that you will make: 1) the velocity of a drop with
no field and 2) the velocity of the same drop in the presence of an electric field. To find the velocity you must time
how 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 a
single voltage to apply to the parallel plates for all of your measurements (between ∼100 and 500 volts) and record
this value as measured on the voltmeter on the front of the apparatus. Next you want to isolate a single drop inside
the apparatus to perform detailed measurements on. Record several (∼10) velocities for this drop with and without
the electric field applied. Find the average velocities and calculate the charge on this drop before measuring any
more. You want to measure drops that only have one or a few charges on them. So, based on your measurement of
the 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 much
volume - short hard squeezes should work better than slower ones.
42
CHAPTER 8. MILLIKAN EXPERIMENT
• If no drops are going in the apparatus check to make sure the hole in the top plate is not clogged with oil. If
it is, blow out the excess oil and try again. IMPORTANT: Make sure the HV is off and disconnected before
taking the apparatus apart.
• The easiest and most accurate way of timing the motion is to start the timer when the drop passes behind one
line 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 the
times 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 of
gravity pulling the drop down and a drag (friction) force. If there is an electric field present the gravitational and
drag forces will still be present but there will also be a force on the drop from the electric field. If all these forces
are 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 field
present 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 electric
field intensity, q is the charge on the drop and v1 and v2 are the terminal velocities for no field and with a field
respectively. 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. To
find the size we will use Stokes’ Law which relates the radius of a spherically shaped object to the velocity at which
it falls in a viscous medium:
r
9ηv1
(8.3)
a=
2gρ
where a is the radius of the moving object, η is the coefficient of viscosity for the medium in which the object is
moving, v1 is the object’s velocity and ρ is the density of the object. In order to apply Stokes’ Law to our oil drops
we must add a correction factor to the viscosity, η, which gives us an effective viscosity of:
!
1
ηeff = η
(8.4)
b
1 + pa
where b is a constant and p is the atmospheric pressure. We now have the necessary equations to calculate the charge
on a drop of oil. First we need to calculate the radius of the drop:
s 2
b
9ηv1
b
a=
+
−
(8.5)
2p
2gρ
2p
We can then find the mass of the oil drop:
m=
4 3
πa ρ
3
(8.6)
8.2.
REAL MILLIKAN EXPERIMENT
Finally we can find the charge of the drop by combining equations 8.1 and 8.2:
mg
v2
q=
1+
E
v1
43
(8.7)
since we know g, we can calculate m, E is the ratio of the voltage between the plates to the distance between them
and 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 in
kg/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 you
did 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 and
show 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?
44
CHAPTER 8. MILLIKAN EXPERIMENT
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 a
photon whose energy corresponds to the energy difference between the excited and ground states. Typically, this
energy difference is in the 100 keV - 3 MeV range.
M4
60
66
1.6
≈ 137
55Cs
0
Qβ−=1175.63
94.4%
5.6%
9.61
12.1
11/2–
3/2+
.1
7/2+
85
30.07 y
661.660
137
56Ba
0
2.552 m
stable
−
137
Figure 9.1: The decay chain shows that 137
55 Cs decays to 56 Ba via beta decay, n → p + e + ν̄e . The 661.66 keV
137
gammas are produced by the subsequent decay of the excited 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
CHAPTER 9. PHYSICS OF GAMMA SPECTROSCOPY
5+
≈
5.2714 y
60
27Co
0
Qβ−=2823.9
99.925%
<0.0022%
0.057%
7.5
>13.3
15.02
4+
2+
2+
0+
2.0
99 ×10 -6
0.0.90
25
0.0 0761173 05
0
34 .23 E4
0.0 11
07 1 2 6.937 E
99
2(+
6
.98
82158.
M3
20
6.0 57
)
6
13
E
M1 2
32
.50
+E
1
2
E2
46
60
28Ni
2505.765 0.30 ps
2158.64 0.59 ps
1332.516 0.713 ps
0
stable
60
−
+
60
Figure 9.2: 60
27 Co decays to 28 Ni via beta decay, n → p + e + ν̄e , and 99.93% of the time to the 4 state of 28 Ni.
+
The 1173 keV gammas are produced by the subsequent decay to the 2 state, and 99.98% of the time the 1333 keV
gammas in the transition from 2+ state to the ground state.
particles, each of which carries an amount of
nucleus. In the second case, the excited states
proceed through β decay, α decay, fission, or
decay processes for two of the commonly used
energy that is equal or larger than the energy needed to excite the
are fed by disintegration of another nucleus. This disintegration may
some combination of these processes. Figures 9.1 and 9.2 show the
sources, 137 Cs and 60 Co.
In the case of 137 Cs, a neutron in the cesium nucleus turns into a proton via beta decay (n → p + e− + ν̄e ), which
changes the nucleus to 137 Ba. 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 ground
state, it releases this 661 keV in the form of a photon.
The case of 60 Co is somewhat more complicated. This nucleus also decays by β decay, and forms 60 Ni 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 in
the ground state. However, rather than decaying directly to this ground state, the nuclear disintegration is a two-step
process in this case. First, the excited nucleus makes a transition to a state located 1333 keV above the ground
state, then it makes the transition to the ground state. In both transitions, the energy difference is carried away by
a γ ray, and therefore a 60 Co source emits equal numbers of γ rays with 1173 keV and 1333 keV, respectively. As
indicated in the decay scheme, there are also some other transitions that play a role in this decay but they are quite
rare.
In the following experiments, we will concern ourselves with the properties of these radioactive sources and the
interaction of γ rays with matter.
9.1.
9.1
CALIBRATION AND ENERGY RESOLUTION OF DETECTOR
47
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 light
when 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 (and
thus the charge carried by the electric pulse) is in principle proportional to the total energy dissipated by the
ionizing particles that created it. The scintillation light produced by the detector is transformed into an electric
pulse by a photomultiplier tube (PMT). The scintillation photons produced in the NaI detector produce electrons
in the photocathode of the PMT, which is optically coupled to the detector, through the photoelectric effect. These
photoelectrons are accelerated in the PMT towards a structure of metal plates called dynodes. Typically, the
potential difference that drives this acceleration is 75 V between each set of consecutive dynodes, and since there are
8 acceleration stages, the PMT operates at ∼ 600 V. At each dynode, the bombardment with accelerated electrons
leads to the emission of more electrons, the number of electrons is thus multiplied at each of the 8 stages. The
resulting 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 of
scintillation photons, at least for events in the photopeak, is proportional to the energy of the γ rays emitted by the
source, the amplitude (or the total integrated charge) of the pulses produced by the PMT is proportional to that
energy.
The objective here is to measure the energies of these gamma rays from various different sources with accuracy. We
first need to establish a procedure through which we convert the channel number (counts) to energy (keV). Begin by
placing the 60 Co source in front of the detector. Switch the power supply on, and slowly increase the high voltage
for the PMT to its nominal value (4.0 on the 10-turn potentiometer). The generation of charge pulses is indicated
by a flickering red light on the unit.
Record the 60 Co spectrum on the computer with the WinDAS program (see manual for details on how to use this
program 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/decrease
the high voltage (i.e. the amplification of the PMT), the two peaks move up/down. Set the high voltage such that
the 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 20
and higher.
Now, we are ready to calibrate the signal distribution, i.e. you will have to establish the relationship between channel
number and γ ray energy. WinDAS has a special program to do this (see manual for details). Although there is
no unique way of accomplishing calibration, let’s pick two known peaks 22 Na (1275 keV) and 57 Co (122 keV) for
this purpose. First record the spectrum of 22 Na. When you have accumulated a nice peak at 1275 keV, replace the
source by 57 Co and see the 122 keV pop up. When you have accumulated enough statistics, you can proceed with
the 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 for
further analyses. Use all of these sources: 22 Na, 40 K, 54 Mn, 57 Co, 60 Co, 109 Cd, 133 Ba and 137 Cs. Look up the decay
schemes for these isotopes in [6] and complete Table 9.1. In completing this table, indicate what kind of a nuclear
process 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 Eimeas measured? How is the statistical precision of this data point determined?
48
CHAPTER 9. PHYSICS OF GAMMA SPECTROSCOPY
Table 9.1: Calibration measurements with γ rays. E1meas refers to the measured peak position of the most dominant
decay where σ(E1meas ) is the measure (standard deviation) of width of the peak. E1acc refers to the accepted value
for γ energy. The subscript 2 refers to the second most frequent decay mode.
Isotope
Decay
E1meas
(keV)
σ(E1meas )
(keV)
E1acc
(keV)
E2meas
(keV)
σ(E2meas )
(keV)
E2acc
(keV)
Activity
(Bq)
22
Na
K
54
Mn
57
Co
60
Co
109
Cd
133
Ba
137
Cs
40
3. Is the detector linear in the measurement range (0 to 1700 keV)? How can you quantify this measure of linearity?
How about plotting (Eimeas − Eiacc )/Eiacc vs Eiacc ? How is this plot useful?
4. What is the energy resolution of this detector? In other words, how precisely does this detector measure the
energies of these γ rays? Plot σ(Eimeas )/Eimeas vs Eimeas . What would be a good functional form to fit these
data 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 background
measurements? 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 electron
is 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 compare
to what is indicated on the labels.
10.
has an energetic (2.505 MeV) gamma emission but it is very rare (2.0 × 10−6 ). Knowing the activity
of the source, the efficiency and geometrical acceptance of the detector, for how long should you take data to
observe a statistically significant peak above background? Substantiate your arguments with data.
60
27 Co
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 statistically
significant data for hematite, carnotite and the unknown sample. Identify the dominant peaks. Estimate the
gamma activity from these rocks. Discuss the natural decay chain. Learn about the chemical composition of
these rocks. What can you say about the unknown sample?
13. Generate backscattered γs by placing the lead cover on the unit. Use the 137 Cs source for this purpose. Record
spectra with and without the cover on (equal times) and subtract the uncovered spectrum from the covered
one. This should reveal the backscattered γs. What is the average energy of these γs? What would you expect
this energy to be?
9.2.
ENERGY LOSS BY GAMMA RAYS
9.2
49
Energy Loss by Gamma Rays
The objective here is to investigate the attenuation of gamma rays through materials. First, we wish to understand
the mechanisms through which a γ loses its energy as it travels in a medium and second, we would like to quantify
the 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 . The
photoelectric 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 photon
leaves the atom neither ionized nor excited. In the case of Compton scattering, as we will study in the following
experiment, the photon scatters off an electron and imparting some of its momentum to the atomic electron. The
production 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. The
Giant Dipole Resonance (σg.d.r. ) is when the target nucleus is broken up and not surprisingly this effect is significant
when the photon energy is in the same order of the nuclear binding energy1 .
(a) Carbon (Z = 6)
- experimental σtot
Cross section (barns / atom)
1 Mb
σp.e.
1 kb
σRayleigh
1b
κ nuc
σCompton
10 mb
κe
(b) Lead (Z = 82)
Cross section (barns / atom)
1 Mb
- experimental σtot
σp.e.
σRayleigh
1 kb
1b
10 mb
10 eV
σg.d.r.
κe
σCompton
1 keV
κ nuc
1 MeV
Photon Energy
1 GeV
100 GeV
Figure 9.3: Photon total cross sections as a function of energy in carbon and lead showing contributions from different
processes (see text for details) [9].
In this experiment, we measure the gammas that are removed from the photopeak by the processes described above
when a piece of lead is placed between the the gamma source and the detector. We use the 662 keV photons from
1 The
data for different elements, compounds and mixtures can be found in http://physics.nist.gov/PhysRefData
50
CHAPTER 9. PHYSICS OF GAMMA SPECTROSCOPY
137
55 Cs
and the NaI(Tl) detector for this study. The decrease of intensity of gamma rays after they have traveled
through 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 in
units of cm2 /g and x represents the density thickness of the material in units if g/cm2 . The density thickness is
simply 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 measurement
period 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 background
subtracted data divided by the live time.
2. From the plot above determine the density thickness of the material that will reduce the original intensity by
half. More specifically,
ln(1/2)
x1/2
= −µx1/2
0.693
=
µ
(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 for
µ. 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. What
is the error in your measurement?
9.3.
9.3
DEEPER LOOK INTO
60
CO SPECTRUM
Deeper Look into
60
51
Co Spectrum
Let’s understand the fundamental features of the 60 Co isotope. The decay chain is shown in Figure 9.2 and a sample
spectrum is given in Figure 9.4.
4
4000
10
3500
3
10
Event/2.85 keV
Event/2.85 keV
3000
2500
2000
1500
2
10
1
1000
10
500
0
0
0
500
1000
1500
2000
Energy (keV)
2500
10
3000
2500
Energy (keV)
2000
1500
1000
500
0
500
1000
1500
2000
Energy (keV)
2500
3000
500
1000
1500
2000
Energy (keV)
2500
3000
1
Percentage Difference From Straight Line
3000
0
0
200
400
600
Energy (Count)
800
1000
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
Figure 9.4: Top left plot is the measured spectrum of 60 Co 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 in
logarithmic scale where a clear third sum-peak is visible. The bottom left curve shows the fitted calibration curve of
the form y = mx + b between the measured counts from the detector to energy (keV) units using the three known
peaks. The bottom right plot displays the precision of this calibration curve where the percentage difference between
the 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 the
highest. 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 below
1000 keV and although not clear from the spectrum there is another shoulder below the 1173 keV photopeak.
These are where an incoming photon from 60 Co undergoes a head-on Compton scattering at θ = 180◦ and transfers
52
CHAPTER 9. PHYSICS OF GAMMA SPECTROSCOPY
maximum energy to an atomic electron in the detector. Since there are two photons with distinct energies from the
source, we get two what is called Compton edges and these correspond to the maximum energy that an electron can
have 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 within
the 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 conservation
of energy and momentum. Check if these correspond to the shoulders or Compton edges.
5. Calculate the Compton edge for gammas at 376 keV (133 Ba), 661 keV (137 Cs) and 1258 keV (22 Na) and compare
them with your measurements.
6. Again from the conservation laws, derive the expression for the energy of the scattered photon from an electron
as a function of the energy of the incoming photon and the scattering angle, θ. This is the Compton scattering.
7. The 60 Co photons are essentially isotropic. So, some photons will backscatter into the detector. At what energy
should 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-positron
pairs are created. If this is the case (as is in 60 Co), we expect to observe two additional peaks in the spectrum
where the positron annihilates with an electron and decays back to two 511 keV photons. If we capture both
of them in the detector, we should observe a peak at 1022 keV, if we catch only one, it should appear at 511
keV. Analyze your data and discuss if this makes sense.
Chapter 10
Physics with Cosmic Muons
10.1
Background Information
Cosmic radiation is high energy electromagnetic waves and energetic charged and neutral particles that are created in
collisions 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 and
neutral pions, and they are short-lived. When at rest, π ± decays into a µ± and a νµ in about 2.6 × 10−8 seconds, or
26 ns. Neutral pions decay into two photons even faster, in ∼ 10−16 seconds. When pions are traveling at relativistic
speeds, because of time dilation as postulated by Einstein and later measured with high precision, they live longer
by a factor γ:
γ=q
1
1−
.
(10.1)
v2
c2
where c is the speed of light and v is the pion’s speed. In terms of their energy and mass, we can recast Equation
10.1:
E
K + mπ c2
γ=
=
.
(10.2)
mπ c2
mπ c2
where K is the kinetic energy of the pion, and mπ is its mass. Thus, the possible values for γ range from 1 (pion at
rest) to perhaps several thousands for highly energetic pions in air showers. We can simply estimate the distance d
traveled by a pion in SI units for γ = 103 .
m
) × (26 × 10−9 s) ≈ 8000 meters
(10.3)
s
Since proton-nucleus interactions occur 20 to 40 km in altitude, pions in general do not reach the Earth’s surface
but instead decay into muons (µ) and neutrinos (ν).
d = γvt ≈ γct = (103 ) × (3 × 108
While neutrinos undergo weak interactions, muons interact electromagnetically. Neutrinos go through the Earth
essentially unnoticed, but the muons start losing their energy through ionization before they decay into a pair of
neutrinos and an electron (µ− → νµ + ν̄e + e− ). Muons, when at rest, live about a hundred times longer compared
to pions: 2.2 × 10−6 seconds, or 2.2 µs. Thus, if they are energetic enough they will travel many kilometers before
they 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 to
charged particles, were used to study cosmic radiation and many of the elementary particles that we know today were
discovered using this detection device. One of the major drawbacks of this approach is that the stack of emulsion
plates are exposed for a long time, thus they integrate many events over time. If one wants to study the cosmic ray
events 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 transparent
medium such as plastic that is doped with fluorescing dye molecules. These fluorescent molecules get excited by
the radiation and subsequently decay (de-excite) by emitting photons in the UV and/or visible region. The photons
bounce 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 incident
on its photocathode. An electron in the photocathode material may absorb the incoming photon from the scintillator
and gain enough energy to leave the photocathode as a photoelectron. The efficiency to liberate such photoelectrons
varies with PMTs but is typically 10 to 20 %. Once the photoelectron is freed it is accelerated toward multiplication
stages (dynodes). Each electron that hits a dynode can liberate several secondary electrons from it’s surface. A
PMT with 12 such multiplication stages has a gain factor G of ∼ 312 , or ∼ 105 . Thus a single photon seen by a PMT
might 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 pulse
for a single photon. Typically thousands of fluorescent photons are created when a charged particle traverses the
scintillator (a 1 MeV electron will produce roughly ten thousand photons in a scintillator like ours) which ultimately
leads 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 the
sockets (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 the
PMTs until the standby light is on, and you flip the second switch to the on position.
10.3.
10.3
COSMIC RAYS IN LUBBOCK
55
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 shows
the expected vertical rate of cosmic particles as a function of altitude. The approximate rate of muons is roughly 70
m−2 s−1 sr−1 at sea-level. The neutrino rate is about twice that of the muons. You may find
http://www2.slac.stanford.edu/vvc/cosmicrays/crslac.html useful. You can also download CORSICA from
http://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. The
points 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 quantitative
information.
4. What is the discriminator threshold for each channel? Test the effects of different levels (be careful with the
trim pots) of thresholds. These levels strongly affect your measurements, explain why.
5. What is the (average) cosmic ray rate (in Lubbock)?
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CHAPTER 10. PHYSICS WITH COSMIC MUONS
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 experimental setup, method and results. Be sure to discuss the statistical and systematic errors (see Figure 10.2 as
an 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 cosmic
ray 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 solid
line is a fit of the form cos2 θ.
10.4.
10.4
ENERGY SPECTRUM OF COSMIC MUONS
57
Energy Spectrum of Cosmic Muons
In this experiment, we will measure the energy spectrum of vertical cosmic muons. Once the cosmic ray counters
are setup in coincidence mode with about a meter between them, the mass in between the counters is incrementally
increased and the cosmic ray rate is measured. In other words, we would like to see how much material it takes to
stop the cosmic muons. Figure 10.3 schematically shows the setup.
Muon
Top Counter
~1 m
Lead Bricks
Support
Structure
Bottom Counter
Floor
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 bricks
one 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 area
of the detectors and you measure the event rate one lead brick layer at a time. What does this tell you about
the energy spectrum of the muons?
2. Do you think there is a difference in the energy spectrum of vertical and horizontal cosmic muons? Why or
why not?
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CHAPTER 10. PHYSICS WITH COSMIC MUONS
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