Modern Physics II Lab Manual University of Puget Sound Spring Semester 2014

Transcription

Modern Physics II Lab Manual
University of Puget Sound
Spring Semester 2014
2
Contents
1 Measurement of e/m for the Electron
1.1 Doing the Experiment . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Analysis and Writeup . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Hooking up the e/m Power Unit . . . . . . . . . . . . . . . . . .
5
7
8
11
2 Millikan Oil Drop Experiment
13
2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Writeup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Spectroscopy
19
3.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Other spectral series . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Photelectric Effect
23
4.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 X-Ray Diffraction
5.1 Procedure . . . . . . . . . . . . . . . . .
5.2 Measuring the Kα and Kβ wavelengths
5.3 Measurement of bond length . . . . . .
5.4 Analysis and Report . . . . . . . . . . .
5.4.1 Peak Angle Determination . . . .
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29
31
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34
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6 Spreadsheet Energy Levels
37
6.1 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.2 The Finite Square Well . . . . . . . . . . . . . . . . . . . . . . . 41
6.3 A Triangular Potential . . . . . . . . . . . . . . . . . . . . . . . . 42
7 Radioactivity
45
7.1 Finding the Optimum GM Tube Voltage . . . . . . . . . . . . . . 46
7.2 Measuring the Background Count Rate . . . . . . . . . . . . . . 47
7.3 Determining the GM Tube Resolving Time . . . . . . . . . . . . 48
3
4
CONTENTS
7.4
7.5
7.6
7.7
Measuring the Half-Life of 137 Ba . . .
Qualitative Properties of α Radiation
Beta Radiation . . . . . . . . . . . . .
Gamma radiation . . . . . . . . . . . .
8 Nuclear Spectroscopy
8.1 Operation and Calibration . .
8.2 Energy Scale and Resolution
8.3 Compton Scattering . . . . .
8.4 Using Gamma Spectroscopy .
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53
55
56
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59
A Review of Error Analysis
61
A.1 The error from the statistical error on the mean . . . . . . . . . . 61
A.2 Error propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 62
A.3 Combining several measurements of the same quantity . . . . . . 64
Laboratory 1
Measurement of e/m for the
Electron
Although we take the electron for granted nowadays, it was not discovered until
the end of the 19th century. During the 1890’s, several people studied cathode
rays in vacuum tubes and gas discharge tubes. Electric fields and magnetic fields
could deflect cathode rays, so it seemed that cathode rays contained rapidly
moving charged particles. The question was what kind of particles they were.
Were they charged atoms (ions) or something else?
Two important experiments helped settle the matter. In 1899, J. J. Thomson
measured e/m, the ratio of the charge to the mass of the particles. And in 1909,
Robert A. Millikan accurately determined e, the charge of the particles (as you
will do in the next lab). Taken together, these two results allow one to calculate
m, the mass of the particles, which turned out to be about 1,800 times smaller
than the mass of the smallest atom, hydrogen. This result confirmed the idea
that these particles are parts of atoms, smaller than the whole. For his role, we
usually credit J. J. Thomson with “discovering the electron.”
In this experiment you will be measuring e/m in an experiment similar to
that of J. J. Thomson. Your apparatus includes a vacuum tube and a set of
coils that produce a magnetic field. Figure 1.1 below shows what goes on inside
the vacuum tube. Practice your right hand rule and see if you agree with the
path shown, given the directions indicated for the velocity and magnetic field.
The apparatus produces electrons from a hot filament, by thermionic emission. Starting with a small thermal velocity, the electrons accelerate toward
a metal plate held at a large positive voltage Va (called the “plate voltage”).
Some of the electrons pass through a hole in the plate and stream out in a wide
beam. These fast beam electrons are then subjected to an adjustable focusing
field, controlled by the “grid voltage.” They then emerge from the hole in the
metal plate you see in the center of the concentric ring pattern. The beam is
visible because of collisions the electrons make with a low background pressure
of gas atoms, which then emit light.
5
6
LABORATORY 1. MEASUREMENT OF E/M FOR THE ELECTRON
B (into page)
r
Va
hot
filament
6.1 V
Figure 1.1: Schematic representation of the apparatus for measuring e/m for the
electron. The magnetic field is uniform, and pointing into the page, everywhere
along the path of the electron which initially moves straight up after being
accelerated by the potential Va .
On entering a magnetic field, the beam of electrons deflects to follow a
circular path of radius r. The radius of this path can be adjusted so that the
electrons hit one of the fluorescent ring patterns on the metal plate. The radii
of these rings are known to be 0.50, 1.0, 1.5, and 2.0 cm.
The kinetic energy each electron gains as it accelerates from the filament
through the voltage rise Va to the plate is eVa :
eVa =
1
mv 2 .
2
(1.1)
There is technically a small voltage drop across the filament (up to 6 Volts)
depending on where the electrons originate from. But we can ignore this effect,
as it averages to zero since the 6 Volts powering the filament is alternating
current. Thus it is correct, on average, to take the filament at zero voltage with
respect to Va .
The magnetic field, B (discussed in more detail below) is perpendicular to
the velocity of the electrons, and causes them to move in a circle of radius r:
evB =
mv 2
.
r
(1.2)
You should convince yourself that you can eliminate v between equations 1.1
and 1.2 to find:
e
2Va
= 2 2 .
(1.3)
m
B r
The two coils of wire surrounding the vacuum tube are called “Helmholtz
coils.” In this arrangement the coils of wire are separated by an amount equal
1.1. DOING THE EXPERIMENT
7
2rmeas
2.54 mm
2ractual
Figure 1.2: Illustration of the offset of the origin of the accelerated beam of
electrons. The electrons accelerated by the potential Va begin bending in a
circle for approximately 2.54 mm before entering into the visible part of the
tube.
to the radius of either coil. This configuration produces a surprisingly uniform
magnetic field in the central region between the coils, given by
NI
8µ0 N I
≈ 8.9918 × 10−7 T · m/A
,
B=√
R
125R
(1.4)
where B is measured in tesla (T), N is the number of turns of wire in either one
of the twin coils, I is the current in amperes flowing through them, and R is the
coil radius in meters. For the CENCO 71267 apparatus N equals 119. You can
measure R directly (but consider where you should measure it!). An ammeter
must be included in the coil supply in order to measure the current. At this
point you will have the magnetic field, B, the radius, r of the path the electrons
travel, and the supply voltage as measured directly by another multimeter. One
final adjustment must be made, as the electrons emerge from the accelerating
plates into the magnetic field region at a position about 2.54 mm below the
center of the circular hole in the circular target plate so the center of curvature
lies slightly below the target plate, as seen in Figure 1.2. Show that the value
of the radius you measure by sight, rmeas , can be adjusted to very nearly the
correct value for this relatively small shift by using1
ractual
2
2
2
≈ (rmeas ) + (0.00254/2)
.
(1.5)
The final working formula for the experiment is then
e
2Va
h
i .
=
2
m
B 2 r2 + (1.27 × 10−3 m)
1.1
(1.6)
Doing the Experiment
Your apparatus has power supplies that provide electricity to power: (1) the
filament, (2) the accelerating voltage, (3) the focusing potential, and (4) the
1 This
disagrees with CENCO’s writeup by a factor 1/4, but it is the correct form
8
LABORATORY 1. MEASUREMENT OF E/M FOR THE ELECTRON
Helmholtz coils. They have been deliberately disconnected (if not, do so yourself) so that you will understand the circuit by wiring it up yourself. A connection diagram is given at the end of this writeup. Before turning on the power
to the unit have your lab instructor check your wiring.
The filament voltage is fixed to about 6.3 VAC for the outermost two taps
under “FILAMENT”; you can check it with the DMM after turning the unit
on). This voltage should not be adjusted (by moving the plugs to different
taps) – if the filament burns out the bulb is lost, and they are very expensive
to replace! The leftmost knob adjusts the plate (accelerating) voltage, Va , and
should be measured by a multimeter. The second knob adjusts the grid, or
focusing voltage, and the rightmost one adjusts the coil current (there should
be an ammeter there, set to 10 A scale, to measure the current for your magnetic
field calculation). Turn the rightmost knob off, to deactivate the magnetic field
at the start. Turn off the lights, and see the slight glow of the filament under the
circular plate. By adjusting the plate voltage (40-80 Volts) and grid voltage you
should see a very thin ray of electrons going straight up inside the tube. Adjust
the grid to focus this beam. Then, you are ready to take measurements. Since
you can vary the accelerating voltage and current to the coil independently you
can obtain e/m points for a variety of values of Va and I.
In order to keep the equipment functioning properly, please
1. DO NOT EXCEED 170 VOLTS ON THE ACCELERATING (“PLATE”)
POWER SUPPLY.
2. DO NOT ALTER THE FILAMENT POWER HOOKUPS. Do not leave
the filament on for more than 20 minutes at a time. This should be enough
time if you plan your measurements.
3. DO NOT EXCEED 5 AMPERES OF COIL CURRENT. Generally, you
will work with less than this. TURN THE MAGNET POWER OFF
WHEN NOT MAKING MEASUREMENTS. The coils heat up quickly
at high currents, and can melt. NEVER LEAVE THE MAGNET ON
MORE THAN SEVERAL MINUTES AT A TIME.
4. WHEN FINISHED TURN ALL CURRENTS AND VOLTAGES TO ZERO
BEFORE TURNING OFF THE INTEGRATED POWER SUPPLY UNIT.
1.2
Analysis and Writeup
This experiment has several potential sources of error that illustrate the difference between two types of experimental error: statistical and systematic.
Statistical errors refer to those which simply come from an inability to make
measurements with perfect precision. For example, measuring the coil radius
involves eyeing a ruler, and measuring r involves making sure you hit a ring
“dead center.” Presumably, the more times you measure these, the more the
average of all your measurements approaches the “right” answer. Systematic
errors do not get smaller the more times you measure something. These can
1.2. ANALYSIS AND WRITEUP
9
be very misleading if not accounted for. For example if the multimeters you
have used to measure current and voltage were simply not calibrated correctly,
no amount of re-measuring or averaging would help to give the right answer,
and so there is an “inaccuracy” which cannot be reduced in size or eliminated
without an alteration of the experimental procedure and/or apparatus. [By the
way: our meters are calibrated!]
Each data point will give you a measured value for e/m using equation 1.6.
You can find the statistical error using the standard deviation determined from
the full data set. Express your result as
e
= (x.xx ± y.yy) x 10−n
m
Is your value consistent with the standard known result? How many standard deviations away from the accepted value of e/m is your mean value? A
rule of thumb is that if your value is more than about two standard deviations
from the accepted value, the disagreement is likely NOT due to STATISTICAL
errors in your measurement, but due to some SYSTEMATIC error(s).
A closer examination of your data set will provide clues about the nature
and magnitude of both statistical and systematic errors. Here are some ways to
look at your data to for possible error sources:
• Plot e/m versus the accelerating voltage Va . Do you notice a trend?
Does this graph tell you anything about your experiment, in particular,
what might be going on when the accelerating voltage is small? Can you
formulate a hypothesis about what might be going on here?
• Plot e/m versus the current I in the Helmholtz coils. Do you notice a
trend? If yes, does this graph tell you anything about your experiment,
in particular, what might be going on when the magnetic field from the
Helmholtz coils is small? Think through the effect of Earth’s magnetic
field on your measurement. The intensity of BEarth is about 5.4 × 10−5 T
in our region, with 20 µT horizontal component (pointing North, of course)
and 50 µT vertical component. What would you do to reduce the effect
of BEarth , in terms of orienting the apparatus?
If either of these plots show a trend, you have identified a source of systematic
error. In this case, do you still think that the most accurate experimental value
you can extract would be found by simply taking the average of ALL of your
data points ? If not, what do you think a better approach might be?
What part of your experiment contributes the greatest STATISTICAL error?
Estimate how large this error is, and the effect it has on the final error for e/m.
A way to approach this is to estimate typical percentage errors in the relevant
experimentally measured quantities, such as B, I and Va , and then propagate
them through the calculation. The estimate you determine here should agree
with the spread in the measured values in your data set. If there is a discrepancy
between the measured spread and the expected spread then that sends you back
to thinking about other possible sources of statistical error.
10 LABORATORY 1. MEASUREMENT OF E/M FOR THE ELECTRON
What part of your experiment contributes the greatest SYSTEMATIC error?
Try to estimate how large this error is, and explain how this source of error might
result in an incorrect value for e/m.
In light of the above (i.e. lack or presence of trends in the data plots), would
you revise your final value for e/m? In the event that systematic errors are
well-identified, the final measurement value can be quoted in the format:
e
= x.xx ± y.yy (stat.) ± z.zz (syst.)
m
where the sizes of the statistical and systematic errors are given separately.
This allows one to see at a glance where the predominant limitations to the
measurement lie – in limited statistics, or in some inherent limitation of the
experimental design.
For your writeup provide a brief description of the apparatus and procedures,
include graphs of your data set as functions of voltage and magnetic field, and
focus most of your attention on a description of your error analysis. Conclude
with your best result, and statement on whether your experiment agrees or
disagrees with the accepted value for the ratio of e/m. In addition, choose one
of the following two problems to include in your writeup:
1. Derive Equation 1.4 for the magnetic field from a pair of Helmholtz coils.
This arrangement is famous, as an inexpensive way to produce a uniform
magnetic field over a reasonably large range. Find the general expression
for the magnetic field of these coils on the axis of symmetry between them,
as a function of distance z away from the center point. Make a plot of
Bz (z) for the region between the coils and include it in your writeup. Find
the first and second derivates, dBz /dz and d2 Bz /dz 2 , and evaluate them
at the middle point, z = 0, and discuss the plot and the derivatives in
your writeup.
2. The mass of the proton is 1.67 × 10−27 kg. What is q/m for a proton?
What would you have to do to this apparatus to in order to do this experiment with protons? Give conjectures that are reasonable, that you might
be able to implement for modest cost.
1.3. HOOKING UP THE E/M POWER UNIT
1.3
11
Hooking up the e/m Power Unit
The CENCO power unit supplies three voltages and the current for the Helmholtz
coil. These are shown in Figure 1.3. Common ground is used for the three voltages.
• FILAMENT SUPPLY - This is a non-adjustable AC voltage, which you
can tap as any one of several settings from 2.1 V to 6.3 V. From left to
right, the steps between the four taps are 2.2 V, 3.2 V, and 1.2 V. We
are tapping all the way across (setting “6”) to get our filament operating
voltage as 6.6 V. NEVER USE MORE THAN THIS IF THE FILAMENT
BURNS OUT THE TUBE IS USELESS. [Other tap settings indicated on
the unit are 4.4 VAC for setting “4” and 5.5 VAC for setting “5.”]
• GRID VOLTAGE - This is a beam focusing voltage that goes from zero up
to about 80 V. Adjust this by sight to give the tightest beam, depending
on the Plate Voltage. Typical values are around 20-30 V.
• PLATE VOLTAGE This is the accelerating voltage for the beam, adjustable from zero up to about 400 V.
• COIL CURRENT This is a high-current supply to power the Helmholtz
coils. It “kicks in” at about 1.5 A, and is adjustable to about 6 A. The
current is best monitored with a multimeter (set to 10 A scale!) for improved accuracy, as the exact value of the magnetic field depends upon it.
USE THE HIGH-END CURRENTS ONLY FOR SHORT DURATIONS!
POWER SUPPLY UNIT
COIL
CURRENT
PLATE
1.5 – 6 A
FILAMENT up to
GRID
(connects to
6.3 VAC 500 VDC ~ 20 VDC back side)
e/m UNIT
filament
common ground
Figure 1.3: Hookup diagram for connecting the CENCON e/m Power Unit to
the coil/tube unit. The various supplies are described in the text.
plate
grid
12 LABORATORY 1. MEASUREMENT OF E/M FOR THE ELECTRON
Laboratory 2
Millikan Oil Drop
Experiment
Equipment
PASCO Millikan experiment
high voltage supply (500 V)
two digital multimeters
banana plug cables
thermometer (to read room temperature)
ringstands, to elevate apparatus
“I have never begun a laboratory with more misgiving...” might be an apt
paraphrase of the opening line of W. S. Maugham’s The Razor’s Edge appropriate for this lab. While it is central to the history of physics, and very simple
in concept, this lab is challenging. If you care about detail, about quality and
about how the measurements in your lab turn out you will do well at this.
The experiment is a close rendition of the one performed by Robert A. Millikan over the course of several years in the beginning of the twentieth century,
to determine the charge of the electron. Youll be using a commercially available unit made by Pasco Scientific (Model AP-8210), which is a great advance
over previous editions of the experiment that sometimes left students (and instructors) very frustrated. In fact the advertisement for this Pasco apparatus
claims “Typically, a careful student can achieve results within 3% or less of
the accepted value.” The key word here, and with so many endeavors in life is
“careful.”
A copy of the essential pages of the Pasco guide for this experiment will be
provided to you (pages 1-9, and 19 and 20), which includes a derivation of the
equations for converting rising and falling velocities into charges on the droplets.
I’ll go through the essential elements of this theory in the next section, but you
should expand on these in your writeup to give a complete derivation. Extra
suggestions for executing the lab are also given below, as well as systematic
error considerations and a discussion of other essential elements to include in
your writeup.
13
14
LABORATORY 2. MILLIKAN OIL DROP EXPERIMENT
IMPORTANT: It is essential that you read the guide material from Pasco
and this handout before coming to do this lab. This will increase your
efficiency in the lab and satisfaction in doing the experiment.
2.1
Theory
The experiment depends on careful measurement of the time it takes for a small
charged droplet of oil to fall under gravity only – no electric field applied, and
the time it takes for the same droplet to rise under the combined effects of
upward electric force and downward gravitational forces. The first case is characterized by the “falling velocity,” vf , and the second by the “rising velocity,”
vr .
Falling velocity gives information about how big the droplet is. According
to Stokes’ Law, for very small droplet velocities (much less than 1 cm/sec) the
drag force due to the viscosity of air is linearly proportional to the velocity, v
and to the droplet radius, a, according to:
Fdrag = 6πη0 av .
(2.1)
In this equation, η0 is called the viscosity coefficient and it has units N·s/m2 .
When the droplet is falling at constant terminal velocity vf , the upward drag
force balances the droplet’s weight so the net force on it equals zero, and we
can say:
mg = 6πη0 avf .
(2.2)
The viscosity coefficient η0 depends on the air temperature according to the plot
given in Appendix A on page 19 of the PASCO writeup. In order to monitor the
temperature (which increases slightly, the longer the light bulb is turned on),
a thermistor is installed in the chamber’s lower plate. Record its resistance at
regular intervals as you do the measurement, so that you can make the proper
adjustment to the viscosity coefficient.
Charged droplets can be made to rise by applying a voltage of the correct
polarity across the two chamber plates (droplets from the atomizer may be
neutral, or positively or negatively charged). An approximately-uniform electric
field of strength E is created, that can move the droplet of charge q upward. A
new terminal velocity when rising, vr , is reached when the sum of the electrical,
gravitational and drag forces is again zero:
0 = qE − mg − 6πη0 avr
or:
qE = mg + 6πη0 avr .
(2.3)
Elimination of the constants 6πη0 a between equations 2.2 and 2.3 gives the
charge on the droplet:
q=
mgd (vf + vr )
mg (vf + vr )
=
,
Evf
vf V
(2.4)
2.1. THEORY
15
where the electric field strength between the upper and lower chamber plates
is related to the distance of separation, d and voltage difference, V between
them by: Ed = V . This simple equation is the root of the measurement. All
quantities on the right-hand side are known, or measured except for the droplet
mass, m. The rest of the derivation involves finding the droplet’s mass from its
falling velocity in equation 2.2 by using the mass density of the oil and Stokes’s
law.
The oil used in this experiment has a fairly uniform density which is listed
3
in the Pasco writeup as ρ = 886 kg/m . However from measuring a couple of
bottles I’ve found instead the value:
3
ρ = (880 ± 5) kg/m .
(2.5)
This also agrees with a value of 881 kg/m3 obtained in followup discussions with
Pasco representative Steve Meschia. You should use the value for ρ given in
equation 2.5 for your analysis, and its error for assessing systematic uncertainty
(discussed further below). The density, mass, and radius of the droplet are
related by m = (4/3)ρπa3 . Combining this with Stokes’s law in equation 2.2
above, we get that the droplet’s radius is related to its density and falling velocity
by:
r
9η0 vf
.
(2.6)
a=
2gρ
Millikan showed that Stokes’ law is not exactly correct, because the droplets in
the experiment are small enough that their size is comparable to the mean
free path between colliding air molecules. This means the viscous drag is
slightly weaker because the droplet sees more empty space between colliding
air molecules. An adjustment is made for this effect by replacing the coefficient
of viscosity, η0 with an “effective” value that depends on the droplet radius a,
and the air pressure p as:
!
1
ηeff = η0
,
(2.7)
b
1 + pa
where b is a correction factor that is experimentally determined and given on
page 2 of the Pasco writeup as: b = 6.17 × 10−4 (cm of Hg)·cm (you will need
to convert this into MKS units). Observe that in this equation, η0 is reduced by
inclusion of the factor b/pa in the denominator. For large droplet sizes, a, this
effect is small. It is important to have a feel for how significant this correction
factor is or, equivalently, how large the droplets are. IN YOUR WRITEUP,
include a plot or give a table of the correction factor for η0 in equation 2.7 as
a function of a over a range of radii representative of the values encountered in
your experiment. Be sure to discuss the approximate range of droplet sizes you
used in your measurement, and to discuss how significant the correction is.
The corrected viscosity value should replace the uncorrected η0 in equation 2.6 for the droplet radius. Once you do this, solve the resulting expression
for the adjusted estimation of the droplet radius. IN YOUR WRITEUP, show
16
that this results in a quadratic equation for a, and explain why the only feasible
root is:
s 2
b
9η0 vf
b
a=
+
−
.
(2.8)
2p
2gρ
2p
Now we have the essential elements for the analysis, that you can incorporate
directly into a spreadsheet:
1. Estimate the droplet radius, a from the falling velocity vf when no electric
field is applied (equation 2.8).
2. Determine the droplet mass from m = (4/3)ρπa3 .
3. Combine the droplet mass with the falling velocity, vf , the rising velocity,
vr , the plate potential difference, V and the separation distance, d, to get
the charge on the droplet according to equation 2.4.
These are the same steps enumerated on page 9 of the PASCO writeup. The
derivation performed on page 2 of their writeup is tedious if arranged into a
single equation as in their equation (10), and it is unnecessarily complicated. I
suggest you set up separate columns in your spreadsheet for each of these three
steps, to convert each pair of values for a droplet’s falling and rising velocities
into charge on the droplet in a sequence of easy-to-follow steps.
2.2
Experiment
When this experiment is done successfully, it both demonstrates charge quantization and also gives a value for the fundamental unit of charge. You should
approach it in the same way: DO NOT ASSUME charge is quantized, rather
focus your efforts on collecting very good charge information for a small number
of droplets (at least ten), and watch the charge quantization evidence happen
of its own accord. Here are some practical suggestions for the setup of the
experiment:
1. Understand the apparatus before attempting to take any measurements.
Make sure it is level. Make sure to record the number of the apparatus
you used, and use the same one for all your measurements.
2. The high voltage should remain set to 500 volts (as read by the digital
multimeter) so that you need only turn on the switch.
WHILE YOU WON’T BE ADJUSTING THE HIGH VOLTAGE HERE
(IT STAYS AT 500 V) BE CAREFUL AROUND IT!
3. The entire apparatus is mounted on ring stands so that you can set it to a
comfortable height for taking readings. BE CAREFUL NOT TO KNOCK
IT OVER, AND BE CAREFUL IF YOU RAISE/LOWER IT NOT TO
2.2. EXPERIMENT
17
DROP IT, AND TO ENSURE THE SET SCREWS ARE FULLY SECURED AT THE END OF ANY ADJUSTMENT. Always check the leveling bubble that is built into the unit, to see that the apparatus is level,
before you start taking measurements.
4. It really helps to follow the suggestions of the guide from PASCO, in
particular regarding initial focus using the wire (you won’t need to do this
much – presumably with several groups using the apparatus it will stay
around optimum focus for the center of the chamber). Once you have the
focus set, remember to put the small “droplet hole cover” into place (see
Figure 5 in the Pasco writeup). The problem here is NOT in getting too
few droplets into the chamber, but in getting too many of them. One
sharp squeeze of the atomizer is usually enough to create many droplets,
followed by a slow squeeze to push enough droplets in through the hole into
the viewing chamber (with the chamber lever moved to “Spray Droplet
Position”). If you see a diffuse haze of light in the chamber that makes
seeing any individual drops difficult this is usually because the chamber
is filled with too many droplets, most of which are out of focus but are
scattering diffuse light into the eyepiece. In this case clear the chamber
(after you have turned the high voltage off) and start over. Once you have
the right number of droplets in the chamber turn the lever to the “OFF”
position.
5. Please make sure to turn everything off when you are finished:
• the high voltage power supply,
• the halogen light bulb (unplug it), and
• the digital multimeters.
Search for droplets that have small net charge on them, by toggling the
polarity control, and seeking slowly-moving droplets. The approximate rise and
fall times for sufficiently small, and singly-charged droplets are about 20–30
seconds over 10 small grid divisions. After the chamber has been filled initially,
you can clear out droplets that are multiply charged by leaving the plate voltage
on for a few seconds (the highly-charged droplets crash into the plates, while
the singly-charged ones haven’t enough time to get there). You can greatly
improve your measurement precision by holding on to one drop for a long time,
and taking several measurements of the fall and rise velocities. In the end you
can take the average of each and use it to calculate the charge. Do this for at
least 10 droplets. Then organize the results into groups in order of ascending
charge value. A histogram is the ideal way to present your charge data, to
illustrate charge quantization, and to show how you extract your final value for
the fundamental charge.
THE GOAL OF THIS LAB IS TO BE ABLE TO SEE EVIDENCE OF
CHARGE QUANTIZATION WITHOUT ASSUMING IT. THIS MEANS YOU
HAVE TO BE CAREFUL ABOUT USING ONLY THE VERY BEST DATA
POINTS IN YOUR PLOT. METICULOUSNESS IS VERY IMPORTANT!
18
Sometimes your droplet changes charge in the middle of a sequence of measurements, due to interactions with cosmic ray radiation. See if you can hold
on to the same droplet and get more measurements with its new charge. Multiple measurements of the rise and fall velocities are crucial for improving the
precision of charge determination.
Do you see evidence of charge quantization in your data? Comment. If so,
see if you can extract the size of the elementary charge unit, e – the charge of
the electron. You can determine a systematic error on this value by using the
error you will have estimated for how well you measured the plate separation,
d, the uncertainty in the oil density in equation 2.5, and any other common
parameters in equation 2.4 above. Some parameters such as oil density, ρ, are
“buried” deep inside the complicated expression for the droplet charge so that
classical error analysis expressions (which require finding a derivative) can be
cumbersome. In such cases an easy way to determine how much a change in the
density changes the determined charge is by manually inserting slightly larger,
and slightly lower values of ρ into some of the calculations in your spreadsheet
and observing how much it changes the charge.
If you do see evidence of charge quantization, you can take the elementary charge determined from each droplet (dividing by 2, 3, etc. as needed for
droplets with charges of 2e, 3e, etc.) and combine into a single value, and give
the statistical error on this value. An important consistency check is obtained
by quoting the values of e obtained by each of these groups, as well as the overall
combined value.
2.3
Writeup
This is a demanding experiment, so you have two weeks to work on it. However,
it is very rewarding if you put in the care and effort to take good measurements.
Think of the ramifications of what you are doing here: a singly-charged droplet
moving inside the viewing chamber shows the effect of the smallest building
block of an atom!
Be sure to include responses to the individual points (“IN YOUR WRITEUP”)
stated above. When you quote a final value for the electron charge, you can
separate the systematic error effects (uncertainty in the charge plate separation,
d, uncertainty in oil density, ρ, temperature uncertainty, etc.) from statistical
error (the natural spread of your values for e). For example,
e = [1.75 ± 0.25 (stat.) ± 0.35 (syst.)] × 10−19 C .
This is useful to show at a glance whether your experiment is “statistics limited”
or “systematics limited” – whether it would pay to take more data, or whether
a refinement or redesign of the experiment is first called for.
Comment on any ways in which you might improve the experiment if you
were to do it again, or were given the task of building a better oil drop experiment. A good source for historical background on this experiment and some of
the controversy surrounding Millikan’s measurement is given in Wikipedia.
Laboratory 3
Spectroscopy
Equipment
Gaertner-Peck Spectrometer
Sodium light source
High-quality diffraction grating, with mount
Spectral tubes, including H, Hg, He and Ne
In this experiment you will be measuring the wavelengths of the spectral series for various gases using a high-precision spectroscopic measuring apparatus.
Spectroscopy has played a central role in the evolution of modern physics, and
is today an indispensable tool for astronomers and physicists. Each individual chemical element produces its own unique “fingerprint,” a series of spectral
lines which are different from any other element. Furthermore these spectral
lines are subject to modifications under different conditions that influence the
electronic energy level structure, or also if the source is moving with respect to
us. In the former case we have phenomena such as the Zeeman effect (spectral
line separation caused by a magnetic field), or shifts due to molecular bonding,
strained crystal lattices, etc. In the latter case Doppler spectroscopy is used
widely in measuring motions of distant stars, and has been used to establish
the expansion of the universe, and the discovery of extrasolar planets. And of
course there is the remarkable observation of the element helium in the Sun’s
light, before it was ever found here on Earth.
You’ll be using a Gaertner-Peck spectrometer that allows a high degree of
resolution between nearby lines in a series. A high-quality diffraction grating
will split input light into spectral components. In this experiment a sodium light
source will be taken as a reference source whose wavelengths we know, and this
light source will be used to precisely determine the ruling spacing of the grating.
This step calibrates your instrument. After this value has been determined
you’ll be measuring the spectral series of hydrogen, mercury and a noble gas
(helium, neon, argon or krypton - your choice). There is a theoretical formula
for predicting the measured wavelengths of the spectral lines of hydrogen, and
you’ll use this formula to determine a value for the Rydberg constant, RH . No
such simple formula exists for other gases, however, and you will be comparing
19
20
LABORATORY 3. SPECTROSCOPY
adjustable
collimating
slit
diffraction
grating
light
source
COLLIMATOR
ARM
PRISM
TABLE
TELESCOPE
ARM
m =1
LEFT
observer
Figure 3.1: An overhead view of the Gaertner-Peck spectrometer. The telescope
arm swings around in a table whose axis coincides with the center of the prism
table, to accurately measure the angular deflections of spectral lines.
your results with the known spectral lines that have been established by more
sophisticated instruments.
A diagram of spectrometer is given in Figure 3.1. Incoming light is collimated
with an adjustable slit control that is vertically oriented, and which allows you
to narrow the slit to give extremely fine lines actually of light. These are focused
by a lens in the collimator arm and passed on to a diffraction grating, which
splits the colors into different series on the right or left sides. A telescope arm
is positioned with fine adjustment controls and a vernier scale to give precise
locations in angle of the spectral components.
The diffraction grating has very fine lines etched onto a transparent surface.
In addition to passing a large portion of the light straight through, it also causes
secondary maxima to appear at angles θ which correspond to the wavelength of
the light and the order of the maximum, n, according to the familiar diffraction
formula:
d sin θ = mλ .
(3.1)
The order of diffraction can be any integer value m that is small enough to give
a value of sin θ in this equation that is less than one. In order to use this relation
we must have an accurate determination of the spacing between the lines, d, on
the grating.
3.1
Calibration
We will take sodium light as a source whose wavelengths are very well known
(these may have been precisely determined, for example, using an interferometer). Sodium has two very strong spectral lines that are usually indiscernible
because they are very close together. With this instrument you should be able
to resolve them easily. Turn on the sodium light. It takes about 10 minutes
or so for it to warm up to the point where all sodium in the low-pressure tube
3.2. HYDROGEN
21
to vaporize, at which point you will see the familiar yellow color that lights
parking lots around the world . The spectral lines are separated by less than
one nanometer, at wavelengths 588.9950 and 589.5924 nm, which we will take
as given.
Make sure the diffraction grating is well centered and that it faces the light
coming from the slit/collimator arm at normal incidence. Make sure that it is
properly fastened down; if it moves while you are making your measurements,
you will have to start all over. Then measure the angles of the two separated
lines, both to the right and the left sides, and for both first and second order
diffraction maxima. The sighting tube for the apparatus has fine crosshairs that
you can use in concert with the slit width adjustment to determine the angles
of the lines very precisely. You should make sure your apparatus has cross hairs
oriented in an “X” pattern, and not a “+” pattern – that latter makes it more
difficult to precisely find the centers of fine lines since the vertical line partly
occludes them (ask for help to adjust them if they are not already set in this
way).
The vernier angular measurement takes a little getting used to. It allows you
to measure angles down to 1 arc minute of precision. Make sure you understand
how this works by asking for guidance if it’s not clear to you - your measurements
depend on it!
These angles will allow you eight separate measurements of the grating spacing, d, given the wavelengths for the sodium lines assumed above (both left and
right sides, and two orders of diffraction with two spectral lines in each one).
Are your values consistent? Determine both your best estimate for d and the
uncertainty on it, which you can find from the standard deviation of your measurements.
3.2
Hydrogen
Because of the brightness of the sodium lamps the first part could be done in
a lighted room. This is not the case with the following measurements, as they
involve tube sources for the light that are far fainter. For hydrogen, mercury and
the noble gas that follow you will have to develop your own working procedure
for how you’ll get the vertical crosshair centered on the line you are measuring.
It may involve periodically turning on the overhead light just long enough to see
where it is with respect to the line, or shining a modest amount of light towards
your viewing tube in order to show up the crosshairs, etc. Find a method that
works best for you. Having your lab partner assist with lighting, writing down
information, etc. will be a great help for this.
The object of this section of the experiment is to measure the Rydberg
constant for hydrogen. Recall that Rydberg was able to show that hydrogen’s
spectral lines could be predicted by the empirical formula
1
1
1
= RH
−
,
(3.2)
λkn
n2
k2
22
LABORATORY 3. SPECTROSCOPY
where each of the observed spectral lines, of wavelength λkn , corresponds to
a pair of integer values, n and k > n, with the Rydberg constant, RH , as a
common conversion factor. As you know from class, spectral lines correspond
to transitions from an energy level k to a lower energy level n (n < k). The
visible lines of hydrogen have the integer n = 2, corresponding to transitions
ending on the second energy level. This can be rearranged to solve for the
Rydberg constant,
2 2 2 2 m
k n
k n
1
=
.
(3.3)
RH =
2
2
2
λkn k − n
d sin θ k − n2
From equation 3.3 RH is seen to depend entirely on the measured quantities d
and θ, assuming the values m, n, and k are unambiguously determined integers.
Thus you can extract several measurements of RH , which you can use to quantify
your experimental error. Make sure you quote very clearly your final value for
RH with its error, and describe how you found your error, and discuss the
consistency of your measurements.
3.3
Other spectral series
Use the spectrometer to measure the wavelengths of the spectral lines of one of
the noble gases. Use as many values as you can get for each line, by using not
just first but also second order measurements (if possible), and both right and
left side measurements. Make sure to give an error estimation for each of these.
Compare your series to established values for the element you chose from tables
such as the CRC Handbook for Chemistry and Physics, or online values, being
sure to cite which reference you used. Note also that the CRC Handbook gives
line intensity information that you can use to compare to what you observe in
your readings.
Laboratory 4
Photelectric Effect
Equipment
PASCO Photoelectric Effect Apparatus, including:
− set of filters and apertures
− Hg light source
− mounting base rail
− photodiode enclosure
− power supply
− current monitor and amplifier
− PASCO experiment guide for photoelectric effect.
Albert Einstein’s elucidation of the photoelectric effect (1905), together with
Max Planck’s theory of blackbody radiation (1901) form the two seminal experiments that heralded the quantum theory. Heinrich Hertz had already observed
in 1887 that, under certain conditions, light shined on metallic surfaces would
result in the ejection of electrons, in amounts sufficient to constitute a measurable “photocurrent.” Philip Lenard carried out studies by 1902 that established
the basic properties of the effect. Some of these properties conflicted with the
predominant understanding of light up to that time – the wave theory. It was
Einstein’s breakthrough view of light coming in quantized packets called photons, with energy that is directly proportional to the frequency, that was able
to fully account for all properties of the effect. What you will be doing in this
laboratory session is not very complicated – however it is important that you
understand the ramifications of the measurements you’ll be taking.
There are three key properties of the photocurrent:
1. When a photocurrent is present, due to shining the “right kind” of light,
it is proportional to the intensity of the light source. The brighter this
light, the stronger the current will be.
2. The photocurrent appears instantly after the light is turned on – there is
no time delay.
3. If the wavelength of the light falls below a certain minimum frequency
then no photocurrent is obtained, regardless of the intensity.
23
24
LABORATORY 4. PHOTELECTRIC EFFECT
The second property, and especially the third, conflict with the wave theory of
light. How can photoelectrons resist being emitted if sufficient power is delivered
to the metal? In fact if the frequency of the light is below the threshold for
emission of a photocurrent, the metal can absorb sufficient energy to melt into
a pool of liquid, and yet no photoelectrons will be observed!
Einstein’s new view of light suggested that, at the atomic level, light was
interacting with the metal atoms on a quantum-by-quantum basis. Each light
quantum, or photon has energy equal to:
E = hν =
hc
,
λ
(4.1)
where ν is the frequency of the light (in oscillations per second, or Hz), λ is its
wavelength, and h is a constant of quantization called Plancks constant, that
you will be measuring in this lab. Einstein considered the electrons of the metal
atoms to have a certain minimal “binding energy,” called the “work function,” φ,
which had to be delivered by an incident photon in order to liberate an electron.
Once this minimum energy is “paid,” anything left over goes into kinetic energy
of the free electron.
Consider a metal sample that has been curved into a semicircular “chute”
which is concentric with a “collector” wire that gathers electrons incident upon
it, as illustrated in Figure 4.1. Light that is incident on this chute will eject
electrons in all directions with respect to the inside surface. If an electron
happens to emerge at nearly normal incidence there is a good chance it will
make it to the center rod, where it contributes to a measurable photocurrent.
If a potential difference is applied between the rod and the metal chute in
such a polarity as to deter the electrons from the rod, only those electrons
that are sufficiently energetic and that emerge from the chute surface heading
almost exactly for the rod will make it there. And if the potential difference
becomes too large, no electrons will have enough energy to register a current.
Recall that a charge q accelerated through potential difference ∆V acquires
kinetic energy K = q∆V , so the kinetic energy imparted to the photoelectrons
by light of a single frequency can be measured by the maximum amount of
opposing potential difference the photoelectrons are capable of overcoming as
they move from the atom of the chute that they have been liberated from, to the
“collector” rod, or anode.1 In this experiment you’ll be measuring this maximum
opposing potential, or “stopping voltage,” V0 , as the amount of reverse-bias
voltage that just causes the current in the photocell to turn off. This is a
challenging measurement because the current can drop to extremely small levels
(pico-Amperes) before it truly goes to zero. After all, as the stopping potential
is approached it takes a very “lucky” electron - one that heads EXACTLY from
the cathode to the anode - to overcome the reverse-bias potential.
1 The term “anode” refers to an electrode or a surface from which conventional current flows,
and “cathode” refers to the electrode/surface that conventional current flows to. Because
the electrons are negatively charged this makes the rod the anode. It also explains why,
historically, beams of electrons were once referred to as “cathode rays” since they always
emerge from the cathode.
4.1. PROCEDURE
25
incident light
V
–
cathode
+
A
anode
(collector)
photoelectron
Figure 4.1: The phototube consists of a metal chute (cathode) from which
electrons are ejected by the photoelectric effect, and a collector rod (anode)
that is concentric with the chute. A reverse-bias voltage ∆V is applied that
repels electrons from the rod. The number of electrons per unit time that
nonetheless manage to reach the collector comprise the photocurrent measured
by the ammeter, A.
The photoelectric effect can be described by the Einstein equation, which
expresses the kinetic energy of the liberated electron at the stopping potential
as the difference between incoming photon energy and the work function:
eV0 = hν − φ ,
(4.2)
where e is the charge of the electron. Since the frequency and wavelength of
light are related by c = νλ, this can be rearranged as:
hc 1
φ
−
.
(4.3)
V0 =
e λ
e
This equation suggests the overall strategy of this lab: you can measure the
work function, φ, and also Planck’s constant, h (taking the charge of the electron and the speed as light as given) by measuring the stopping voltage as a
function of inverse wavelength, for several different wavelengths. The sources
for these wavelengths will be a low-pressure mercury vapor lamp, which emits
prominently at five different wavelengths (including one ultraviolet line). Color
filters are used to select individual wavelengths.
4.1
Procedure
The self-contained apparatus comes with a mercury light source mounted on a
rail and a receptor unit which has a built-in phototube containing a rounded
chute (cathode) and a rod that runs down the center of the chute that collects the
photocurrent (anode). Power supply and current monitoring units are connected
26
as described in the PASCO guide (note well the color coded cables). The current
monitor greatly amplifies the photocurrent, while introducing relatively little
noise – at its most sensitive setting the output reading is in units of 0.1 pA
(10−13 A)! The apparatus allows you to apply voltages that are either reversebiased or forward biased.
The three investigations you will carry out are outlined in detail in the
PASCO guide, on pages 9–19:
1. Measurement of h from the dependence of V0 vs. 1/λ. You should do this
for each one of the three different apertures.
2. Measurement of current vs. voltage for same frequency, different intensities (by using different-sized apertures). You can do this for any one of the
wavelengths of mercury light given in Table 4.1 that you choose (not just
the 436 nm line suggested in the PASCO writeup), using three different
aperture sizes.
3. Measurement of current vs. voltage for different frequencies at constant
intensity. Use the intermediate (4 mm) aperture and take data for the
three visible lines of mercury: blue, green and orange.
Mercury produces a variety of spectral lines, including several lines that lie
very close to each other (see, e.g., C. J. Sansonetti, M. L. Salit and J. Reader
in Applied Optics 35 (1996) 74-77; or the CRC Handbook of Chemistry and
Physics). Two of these are in the ultraviolet range - bleached white copier paper
can reveal these lines, but the filters that come with the apparatus are very well
attuned to these wavelengths. For our purposes well take the wavelengths of
mercury as being in five separate groups:
line label
UV 1
UV 2
blue
green
orange
wavelength (nm)
365.22
404.93
435.74
546.1
578.15
2
Combination of four lines at 365.016 nm (relative intensity = 2800), 365.484 nm
(300), 366.289 nm (80) and 366.328 nm (240).
3
Comb. of two lines at 404.657 nm (rel. int. = 1800) and 407.784 nm (150).
4
5
Table 4.1: Dominant spectral lines of mercury in the visible light range.
4.2. ANALYSIS
4.2
27
Analysis
Determine Planck’s constant from the slope of a linear fit through your data
points from Experiment 1, and show the fitted line on a plot of V0 versus (1/λ).
You can incorporate individual errors on V0 at each wavelength (which may differ
from each other) by performing a weighted linear least–squares fit (formulas for
this can be found in the appendix). This fit will provide you with the best-fit
slope and intercept values and errors on those values. Or, you can estimate the
error in the slope by eye, drawing the line of highest possible slope through the
error bars on your data set, as well as the one of lowest slope. This is clearly
not precise, so be conservative - it is of course much easier. The slope and
intercept, and the errors on them, translate directly into values and errors for
h and φ. The value of φ you obtain will most likely not correspond to the work
function of any known metal. Its value will be too small because when extremely
small currents are measured in an electrical circuit, there is some “interference”
that occurs at every junction point between two metals that have different work
functions (including wires, circuit traces - everything!) These differences in work
functions are called “contact potentials”, or “Galvani potential differences” and
they directly affect the measured value of stopping voltage. In order to ascertain
the net contribution of the contact potentials in our circuit we would have to
do more investigative work that is beyond the scope of this laboratory.
Determine a value of h, and its error, from the data taken with each of the
three apertures. Discuss the consistency of your values of h with the established
value, 6.63 × 10−34 J · s. Is there any bias or inconsistency in your result? If so,
do you understand what could have caused it? It is nice to put these on a plot
of h versus aperture size, including a horizontal dashed line for the standard
value of h. Take care in quoting values and errors. A common offense in dealing
with errors is to cite too much precision. E.g., the statement
h = 6.799 × 10−34 ± 5.99421 × 10−35 J · s
NOT GOOD
is 1) hard to read, 2) has differing precision between central value and error,
and 3) contains excess significant figures. Use the error to guide you in the
number of significant figures quoted (keep AT MOST two significant figures in
the error), and write the final result in a way that makes the answer, and the
error as clear and easy to interpret as possible:
h = (6.80 ± 0.60) × 10−35 J · s
MUCH BETTER
For Experiment 2 include a single plot showing the current vs. voltage for
each one of the three apertures, 2 mm, 4 mm and 8 mm. Discuss the behavior
in this plot, comparing the similarities and differences in the three series. How
is this behavior seen in the plot consistent/inconsistent with the wave theory
of light? How is it consistent with the photon theory? You may want to “blow
up” the area around the convergence point to address this.
For Experiments 3 include a single plot of showing the current vs. voltage
for each one of the visible wavelengths from Table 4.1. Discuss the behavior
28
you see in this plot, including which aspect(s) cannot be explained by the wave
theory of light, and how the photon theory succeeds in explaining them.
Keep in mind the significance of what this experiment establishes. It is a
fundamental paradigm shift in our picture of the universe: light interacts with
matter with discrete, not continuous, exchanges of energy and momentum. This
is the essence of quantum electrodynamics, one of the most successful physical
theories of the 20th century.
Laboratory 5
X-Ray Diffraction
Equipment
Tel-X-ometer 580M X-ray diffraction apparatus
Tel-X 2590 Driver Motor, Digicounter and Ratemeter units
Computer interface unit, tube current monitor cables
Computer with software installed
The planes of atoms in highly-symmetric crystals can diffract electromagnetic radiation of wavelengths comparable to the spacings between planes of
atoms in the crystal lattice. Ionic-bonded crystals such as NaCl, KCl, RbCl
and LiF have very simple lattice structures with lattice spacings on the order of
a fraction of nanometer between planes of atoms. This makes them very suitable
for diffracting X-rays with wavelengths of 100-300 pm. The Bragg diffraction
law says that if an X-ray of wavelength λ is incident at angle θ on a crystal with
layers of spacing d as shown in Figure 5.1, then diffraction maxima result if the
path length difference equals an integer number of wavelengths:
nλ = 2d sin θ .
(5.1)
θ
θ
d
Figure 5.1: Bragg scattering in a regular crystal, with d equal to the distance
between successive planes of ions.
29
30
LABORATORY 5. X-RAY DIFFRACTION
Make sure you understand the derivation of this result, by examining the path
length difference between the two rays shown and seeing that the equation
simply requires this difference to be an integral number of wavelengths. The
index, n, gives the order of the diffraction maximum: n = 1 is a path difference
of one wavelength, n = 2 for two wavelengths path difference, and so on.
The goal of this laboratory is to achieve familiarity with the production and
use of X-rays by measuring the lattice spacings in ionically-bonded crystals from
the Bragg scattering peaks. It will be very reminiscent of your experience with
optical spectroscopy, but with peaks in the X-ray region instead of the visible
light range.
Familiarize yourself with the Tel-X-ometer 580M apparatus. Observe that it
has an interlock device that allows X-rays to be created only with the protective
cover engaged and locked into the center position. The X-rays cannot penetrate
the protective cover. Inside the cover is an X-ray production tube. This tube
has a heated filament that liberates electrons by thermionic emission, and also
lights up when in use. The liberated electrons are accelerated by a 20 kV or
30 kV potential (selectable by the red switch on the top of the unit). X-rays
are created by bombarding a piece of copper with the accelerated electrons.
The X-ray tube’s envelope is made of lead glass, that allows visible light to pass
through but not X-rays, except for a “port” on the front of the tube. X-rays pass
through this port to a target mounted in the center of the rotating measuring
apparatus. Photons of specific energies and wavelengths result when electrons
on the innermost shells of the copper atoms are dislodged by the bombarding
electrons, and the vacancies in those shells are filled by outer-shell electrons
falling into their place. Usually these will be n = 2 electrons falling into n = 1
holes (Kα X-rays) or n = 3 electrons falling into the n = 1 holes (Kβ X-rays).
Which one will be of higher energy and shorter wavelength? Which one do you
expect to occur more often? There are more series, such as the L, M , etc. series
of transitions resulting when n = 2, n = 3 etc. electrons are dislodged and outer
shell electrons fall into their place. But their energies are much lower, and the
resulting wavelengths are too long for crystallography studies (for example, the
Lα line has energy 0.93 keV, or wavelength 1.3 nm, about ten times the Kα
wavelength).
Observe the gearing of the center-mounted chamfered crystal mounting post,
with respect to that of the measuring arm that holds the Geiger Muller tube:
it turns exactly half the angular distance of the measurement arm. Thus the
angular displacements you read around the perimeter of the table are 2θ, where
θ is the angle of either incoming or outgoing X-rays with respect to the crystal
surface as shown in Figure 5.1. Make sure you understand how the geometry of
this figure relates to the X-ray source, the crystal, and the measuring unit of the
Tel-X-ometer. The exercise of determining the spacing between planes of atoms
in a crystal is then one of finding “peaks” in the rate of diffracted intensity, as
a function of angular position with respect to the crystal face.
The Geiger-Muller tube is a standard device for measuring radiation in the
form of either high-energy photons or energetic charged particles. It consists
of a fine wire centered inside a metal cylinder with a suitable gas between
5.1. PROCEDURE
31
them (various mixtures are used, usually an inert gas such as a argon with a
small amount of a hydrocarbon such as methane). A high voltage difference
is applied between the fine wire and cylinder, that creates a high electric field
strength between them. The passage of an ionizing photon or charged particle
initiates an “avalanche” process – an electron is dislodged from an atom of gas
by a penetrating photon or charged particle, and is accelerated away from its
parent ion strongly enough that it gains sufficient energy to dislodge secondary
electrons from other gas molecules, which in turn liberate tertiary electrons,
etc. The “gain” on these tubes is typically in the 104 range (the number of
electrons liberated and collected at the anode wire as the result of a single
initial ionization). The collected charge registers as a pulse/count, and in a
short time (< 1 ms) the tube is “cleared” and ready to register another “hit.”
The GM tube is powered by the Tel-X-Driver unit. The single coaxial cable
running to the tube both electrifies it (the voltage used for these GM tubes is
around 425 V) and also reads out the pulses registered by avalanche activity in
the tube. The GM tube should NOT be powered by voltage higher then 500 V.
The count rates and angles were recorded by hand in past years. However,
in this laboratory we will use the Tel-X-Driver unit to record data. The driver
unit controls a stepping motor, and is under computer control so that it can
complete scans of rate activity, versus angle, in step as small as 1/6th of a
degree. Additionally, the software controlling the stepper unit controls the GM
tube high voltage and readout, and also monitors (and records) the X-ray tube
current.
5.1
Procedure
This section describes the procedure for checking alignment and proper function
of the X-ray apparatus. It may not be necessary to make any adjustments, if
the equipment was recently used or previously aligned.
First check for alignment of the measuring arm with respect to the center
post. Lift the hood up and, after ensuring the stepper motor apparatus is not
engaged with the central gear, move the measuring arm to zero degrees. With
all collimators and GM tube removed from the measuring arm, sight through
the measuring arm to check that the center post is aligned straight with respect
to the X-ray tube. The apparatus has rulings on the center table base that
should line up approximately with the 90 degree marks on both sides next to
them. If you see misalignment, make sure to ask Marcus or myself for assistance
in correcting it.
Crystals of several different types can be mounted into the center post for
study. All crystals used in this study are strongly-bonded ionic compounds such
as NaCl (“halite”), LiF, RbCl, etc., and all of them have color-painted ends for
identification. Please take care with these crystals - they are carefully grown
and cleaved in order to provide pure, well-formed crystal samples with clean
faces for very clear X-ray diffraction peaks. The crystal are stored in plastic
cases with a desiccant to maintain a state of dryness. It is important to return
32
the crystals to their cases when they are not in use.
You will use a LiF cyrstal to check the basic alignment and function of the
unit. Mount a LiF crystal into the center post so its matte (non-shiny) side is
irradiated, and make sure it is secured but DO NOT OVER TIGHTEN. Install
two collimators in the measuring arm: a 3 mm slot collimator at Experimental
Station 13 (first slot) and a 1 mm collimator at ES 22. Rotate the measuring
arm to zero degrees (the GM tube is not yet in place). With the hood still up,
turn the key on and turn the timer knob so the filament heats and lights up
(no X-rays can be produced when the hood is up). Sight through the 1 mm
collimator and make sure the crystal is well-centered with respect to the copper
surface shining light through the 1 mm collimator at the basic port. Once it is,
you can insert the GM tube in ES 30. It may fit snugly against the collimator
in ES 18, so insert gently. Close the hood and engage the interlock.
Start the TEL-X-Driver control software (use Tel x driver usb v2.43.exe).
Note the carriage arm angle position and compare it to the value in the software
interface shown in the upper left hand corner. If the angles do not match, open
the “Setup” window (button in the lower right hand corner), change the angle
in the “Arm position” section to reflect the actual angle of the carriage, and
press “Set arm pos.” Push the red button to start the X-rays (the red “Xrays ON” indicator under the hood lights up). Check the X-ray tube current
in the software interface. It should read 80 − 90 µA. The GM tube voltage is
controlled by the softeware interface, and should be left at 425-430 V. Turn on
the high voltage before proceeding by selecting the green colored HV button in
the software.
The diffraction peak of LIF with Cu Kα radiation is known to occur at 45.0◦ .
Set the Regions of interest (ROI) to some values in this range (such as 38◦ to
50◦ ). Set the Time per step value at .1 seconds. This is a good value for rough
locating of peaks, but a longer time, on the order of 1 second, will yield higher
quality data. Also, select a resolution. A resolution of .1◦ should be sufficent to
view the peaks of LiF. However, as with the time per step, greater resolution
yields higher quality data. The scan is started by clicking the button in the
lower left-hand corner of the computer interface. A graph should be obtained
that has its highest peak within 1◦ of 45◦ . It should look similar to Figure 6.2.
5.2
Measuring the Kα and Kβ wavelengths
NaCl has face-centered cubic crystal structure without any one-to-one assignment of any ion Na+ to a particular Cl− ion. Instead each ion is equally paired
to the six oppositely-charged ions on orthogonal axes around it. So if the density
of salt is known then the bond length follows directly (there is no possibility for
extra space between as this is in a covalent molecular solid). Derive this bond
length, using a precise value for the density of crystalline salt, or halite (see if
you can find it to at least four significant figures). You should get something
close to 0.282 nm. This known bond spacing, and the regularity of the salt
crystal allow it to be used as a diffraction grating to find the wavelength.
5.3. MEASUREMENT OF BOND LENGTH
33
Figure 5.2: Software image of diffraction from LiF showing two strong diffraction
peaks.
With the same set of collimators in place make a complete scan of the diffraction intensity as a function of 2θ, going from 20 degrees up to about 120 degrees.
Try to locate three orders of diffraction maxima. The angles are very narrowly
defined, and you should be able to see Kα and Kβ lines separately, even for
n = 1.
Extract a peak position and uncertainty for each of diffraction maximum. A
suggestion for how to do this is given below. Once you have these use the Bragg
formula to find the corresponding wavelengths, making sure have the correct
order index, m, included. You will have two, and possibly three independent
measurements of each of the copper Kα and Kβ wavelengths with their uncertainties. You can combine each of these sets to quote your final single result
for these two wavelengths, with your uncertainty (by error propagation). Make
sure you include all this information in a table in your writeup.
5.3
Measurement of bond length
Reversing the philosophy of your first measurement, now assume the wavelengths of the copper peaks are λα = 0.154 nm and λβ = 0.138 nm.1 Repeat
1 The exact picture, important for very high resolution measurements, is more intricate:
the Kα line consists of Kα1 and Kα2 lines separated by about 0.4 pm, and the value of 154 pm
is an average of these two. Similarly there are several lines for the Kβ series. The different
34
the scan procedure to determine the peak locations for one of the other ionic
crystals – KCl, RbCl or LiF. These are in labeled tubes and are also color coded.
Please make sure to put them back in their tubes and into the box when you
are finished. In this part you assume known values for the copper X-ray wavelengths and use the Bragg relation to deduce the bond length and its uncertainty
in these salts.
5.4
Analysis and Report
In your report give a basic discussion of the apparatus and theory of operation
for the experiment, and include plots of the count rate data as a function of
angle. Discuss how you analyzed the data, and how you determined the errors
on the final quoted values in each part. In addition, carefully analyze the angular
position of a diffraction peak using the following methodology.
5.4.1
Peak Angle Determination
Suppose that you have identified a peak somewhere by several successive angular
readings of diffraction intensity versus angle, and you want to extract a precise
angular value for the peak together with an error for that value, as shown in
Figure 5.3. In this case the counting rate is shown as a function of location x.
45
40
35
Count rate, r
30
25
20
15
10
5
0
3
3.5
4
4.5
5
5.5
6
Position, x
Figure 5.3: Hypothetical data and counting uncertainties for a peak in counting
rate.
One way to get an estimate for the peak position is to fit a quadratic (you will
need at least three points to do this) in EXCEL (TRENDLINE option). From
energies result from a small splitting of the energy degeneracy between s, p, d, etc. levels in
higher-n shells that occurs in many-electron atoms.
5.4. ANALYSIS AND REPORT
35
the values of the parameters in this fit you can quickly get a peak location.
However due to noise, etc. the peak isn’t particularly well fit by a quadratic,
or even a Gaussian distribution. A simple ad hoc method is to do a weighted
approximation of rate vs. angle for the peak location. Suppose that the locations
are given by xi and that the rate counts at these positions are ri . As usual with
random events occuring at a consistent average rate, we’ll assume the count rates
follow a Poisson distribution, with uncertainty equal to the standard deviation
√
which is the square root of the count rate, ri . We can fit for a peak position
by minimizing the χ2 function defined by:
χ2 =
2
X [ri (xi − x
X
¯)]
2
=
ri (xi − x
¯) ,
√ 2
ri
(5.2)
where x
¯ will be the best estimate of the central location position. Proceeding
by setting dχ2 /d¯
x = 0 gives the result:
P
ri xi
x
¯= P
.
(5.3)
ri
From this the χ2 value at the best-fit peak location x
¯ can be found, and the
uncertainty on this angle can be taken through either standard error propagation
applied to equation 5.2 or by finding the variation in angle from x
¯ that increases
χ2 by 1 unit from its minimum.
36
Laboratory 6
Spreadsheet Energy Levels
The goal of this numerical exercise is to get a better feel for wave functions
and the Schr¨
odinger equation, for how quantization arises whenever a particle
is confined in a potential, and for how the energy level spacing depends on the
shape of the potential. You will be finding the eigenfunction solutions, and their
eigenvalues (the bound state energies) for three different potentials. This lab
is an EXCEL exercise, and you should work together with your lab partner to
complete all the investigations, to discuss and understand your results and to
put all the information together in your writeup.
The Schr¨
odinger equation for stationary state solutions in one dimension is
a second order differential equation in the position, x:
−
¯ 2 d2 ψ
h
+ U (x)ψ(x) = Eψ(x) .
2m dx2
(6.1)
Here, ψ(x) is the wave function representing the particle of mass m immersed in
a potential energy function U (x) and E, which is a number and not a function
of x, is the particle’s energy. A bound-state particle is confined mostly to the
region of space where E > U (x) (unbound particles have E > U (x) everywhere).
In this case the differential equation assumes the form:
d2 ψ
∝ −ψ(x) .
(6.2)
dx2
We already have experience with a differential equation like this, from our work
with harmonic oscillators, and so we know that the solution has to be oscillatory. In regions where E < U (x), the “—” sign in equation 6.2 disappears,
giving an exponential solution instead. We can understand this behavior mathematically by recalling that the curvature of a function f (x) is related to the
second derivative by
f 00 (x)
κ=
(6.3)
3/2
(1 + [f 0 (x)]2 )
(the curvature is equal to the inverse of the radius of curvature). The sign of
the curvature is determined purely by the sign of the second derivative. If the
37
38
LABORATORY 6. SPREADSHEET ENERGY LEVELS
curvature is positive, the curve has increasing slope values (like a parabola that
opens upward), and if it is negative the curvature is like that of a parabola
opening downward. What the Schrödinger equation in the form of equation 6.2
is telling us, then, is that if ψ(x) is positive (above the axis) then the curve heads
downward, back towards the axis and if ψ(x) is negative (below the axis) then
the curve heads upward, again back toward the axis. This tendency to head
back to the axis, regardless of which side of it the function is on, is the root of
oscillatory behavior – it explains why the bound state solutions are wave-like
regardless of the exact form of the potential function.
The three binding potential energy functions you will investigate in this
exercise are:
1. A harmonic oscillator potential, U (x) = 21 κx2 , with spring constant κ =
13, 456.665 N/m;
2. An finite square well of width L = 6.1321295 × 10−10 m, that is exactly
30.000000 eV deep; and
3. A triangular potential, U (x) = a|x|, with a = 2.6784462 × 10−6 J/m.
The reason for the high precision is discussed more below. You will be considering an electron bound within each of these potentials. You already know the
energy levels and wave functions that the Schrödinger equation gives for the
first one, so it will serve as a “test drive” for your numerical method of solution.
The second potential was discussed in class, but we never calculated any actual
wave functions or exact energies for it. You’ll do that in this exercise, for a
specific example of the square well. The last potential is one for which we don’t
have solutions, but you’ll find them out numerically, using EXCEL. Once you
have them you will compare and discuss the energy level structure of these three
potentials for the five lowest-energy solutions in each case.
Since we have EXCEL at our command we may as well use high-precision
CODATA1 (2008) values for physical constants:
m = 9.109 382 15(45) × 10−31 kg
h = 6.626 068 96(33) × 10−34 J · s
e = 1.602 176 487(40) × 10−19 C
¯h = 1.054 571 628(53) × 10−34 J · s
The last two digits, in parentheses, give the uncertainties on the values in the
last two places of the quoted values. E.g.,
m = (9.109 382 15 ± 0.000 000 45) × 10−31 kg .
Start by setting up the numerical solution for the harmonic oscillator potential,
which is one for which we already know the solutions.
1 The Committee on Data for Science and Technology, of the National Institute of Standards
(NIST). See: http://physics.nist.gov/cuu/Constants/.
6.1. HARMONIC OSCILLATOR
6.1
39
Harmonic Oscillator
The equation for the wave function is
−
¯ 2 d2 ψ 1 2
h
+ κx ψ(x) = Eψ(x) .
2m dx2
2
(6.4)
The greek letter kappa, κ, is used for the spring constant instead of k so as not to
be confused with the wave number. For this example use κ = 13, 456.665 N/m.
You have seen the mathematical solution of this problem in your text. First, a
variable transformation was made:
r
r
2πmν
mω
κ
x=
x , where ω 2 =
.
(6.5)
y≡
h
¯
¯h
m
This transformation scales from x values that have very small numerical values
to the dimensionless variable y which has values that are not so tiny. The
differential equation 6.4 then becomes
ψyy + α − y 2 ψ = 0 ,
(6.6)
2
d ψ
2E
where α ≡ 2E
hν = h
¯ ω . The symbol ψyy = dy 2 is a common shorthand notation
for derivatives.
Since the potential energy function is symmetric about y = 0 the wave
function solutions for this equation must be either symmetric (even functions
in y, ψ(−y) = ψ(y)) or antisymmetric (odd functions: ψ(−y) = −ψ(y)). We’ll
make use of this fact by working only with the region y > 0 (you can just reflect
the solution into the region y < 0 to get the rest of the wave function). The
variable y automatically changes x (which is on the order of 10−10 m) to a more
convenient (and dimensionless) distance scale, as the spring constant and mass
involved give:
y = 3.240176 × 1010 m−1 x .
(6.7)
Furthermore if we agree to measure the energy in eV, then with the given spring
constant, which gives h
¯ ω = 1.2817412 × 10−17 J = 80.000000eV, we can put
equation 6.6 into the form:
1
2
ψyy +
ε−y ψ =0 ,
(6.8)
40
where ε replaces α (which contains the energy E) in order to remind us that
it is the energy measured directly in eV, so that the dimensionless quantity
2E
1
α = 2E
hν ⇒ 80 eV ⇒ 40 ε.
As we have discussed, when a particle is bounded by a potential this differential equation has finite solutions for only a discrete set of energy eigenvalues, ε.
However, there is nothing like seeing this for yourself by setting up a spreadsheet
solution.
Now set up a spreadsheet to generate solutions for ψ(y) from equation 6.8.
Start your solution at y = 0, and propagate the wave function and its derivatives
40
(this means at least four columns, for y, ψ(y), ψy (y), and ψyy (y)) in short steps
of about ∆y = 0.01 or so in y, up to y value of about 5, according to the
propagation equations:
1
2
ψyy (y) =
y − ε ψ(y) ,
(6.9)
40
ψy (y + ∆y) = ψy (y) + ψyy (y)∆y , and
(6.10)
1
2
ψ (y + ∆y) = ψ(y) + ψy (y)∆y + ψyy (y) (∆y) .
(6.11)
2
Make sure that you understand the propagation strategy here: for any given
value of y, the second derivative is determined FOR THE CURRENT y VALUE
by the first equation; this second derivative plus the first derivative value FOR
THE CURRENT y VALUE determine the first derivative FOR THE NEXT
y VALUE according to the second equation; and lastly the first and second
derivatives FOR THE CURRENT y VALUE plus the current ψ values determine
ψ FOR THE NEXT y VALUE according to the last equation. If you think
through the logic of this for a minute you will see that two pieces of information
are missing, in this “zipper” strategy for propagating y, ψ, and its derivatives:
the values of ψ and ψy FOR THE CURRENT y VALUE (look closely at the left
hand side arguments in equations 6.9, 6.10 and 6.11). These must be input –
they are the initial conditions, or constants of integration.
It will be convenient to leave separate cells in the spreadsheet for the input
energy, ε and for the starting values, the fundamental constants, and for ψ(0)
and ψy (0) of the wave function at y = 0 so that you can change these easily.
The last two will be adjusted to generate the symmetric and antisymmetric
solutions:
SYMMETRIC:
ψ(0) = positive value
ψy (0) = 0
(odd number of “bumps”)
ANITSYMMETRIC:
ψ(0) = 0
ψy (0) = positive value
(even number of “bumps”)
Set up a spread sheet that goes from y = 0 to y = 5 in steps of 0.01, and try
some different energy values and initial starting conditions. Don’t worry about
what exact starting values ψ(0) and ψy (0) you use (I usually just set them to
1) – we’ll do the normalization later. “Shop around” for energy eigenvalues
that give finite solutions and their corresponding wave functions. Remember
the guiding principle: The wave function must go to zero at large y.
To help you find these it’s convenient to set up two small graphs on your
spreadsheet, one for the wave function out to about y = 2, and another that
goes all the way out to the end (y = 5). Find the energy eigenfunctions and
wave functions for the ground state (one bump, symmetric) and the first excited
state (two bumps, antisymmetric). At first you will see a graph where the wave
function takes off wildly to infinity at the upper end. Keep searching if the
6.2. THE FINITE SQUARE WELL
41
divergence at y = 5 changes from positive infinite to negative infinite - you’ve
crossed an eigenvalue! Go back and “home in” on it.
1. Find the energy values of the first four bound states, and compare
them with the predictions of the known solution, En = n + 21 hν.
Notice the acute sensitivity to getting the energy just exactly right – anything
even slightly above or below a “magic values” leads to strongly divergent solutions. In fact to get a solution convergent out to arbitrarily large y you would
have to continue the searching process to infinite precision in the energy eigenvalue! Also notice that, without knowing anything about Hermite polynomials
or the detailed solution given in your text:
ψn (y) =
2mν
h
¯
1/4
n
1/2
(2 n!)
Hn (y)e
−y 2 /2
1
, En = n +
hν ,
2
(6.12)
and even without a priori knowledge that finite solutions exist only for certain
discrete energy values, your numerical method reproduces all of these results.
You can confirm this by overplotting ψ0 (y) and ψ1 (y) from equation 6.12 on
your numerical solutions to see that the shapes agree. You’ll need to scale your
wave function to do this – try adjusting your initial value ψ(0) for the numerical
ψ0 solution, or the initial value ψy (0) for the ψ1 solution, until you can get them
to fit on top of each other.
2. Plot the first two wave functions you found numerically, together
with the corresponding exact solutions from equation 6.12 above.
Scale the former so they fit against the numerical solutions. Do their
shapes agree?
Since we know that we have to normalize the numerical wave functions anyway,
this confirms we have found the same exact solutions.
6.2
The Finite Square Well
The potential for this part will be a square well of width L centered at x = 0:

x < −L/2
 30 eV ,
0,
−L/2 < x < +L/2
V (x) =
(6.13)

30 eV ,
x > L/2
where the width is L = 6.1321295 × 10−10 m. As a point of comparison, and
to motivate the choice of L, first calculate the energy levels (in eV) that you
would expect for an infinitely-deep square well (whose solutions you know) of
this same width. How would you expect the energy levels of the finite well to
differ from these? About how many bound state energy levels would you expect
to be contained in this finite well?
Now set up your spread sheet just as you did in the previous section, and
find them. Make the height of the square well an adjustable parameter in your
spreadsheet. Use an increment size of and start at x = 0, the middle of the
42
well. We can do this again due to the symmetry with respect to the center of
the well, so that we know there are only symmetric and antisymmetric solutions
as with the harmonic oscillator. With a spread sheet that goes from x = 0 to
x = 10−9 m in a thousand steps of 10−12 m, try different energy values and
initial starting conditions, and “shop around” for the energy eigenvalues that
give finite solutions.
Again it’s useful to have two small graphs on your spreadsheet, one for the
wave function out to about x = 4 × 10−10 m (400 steps – to just beyond the
right wall), and another that goes all the way out to the end. Find the energy
eigenfunctions and wave functions for the ground state (one bump, symmetric)
and the first excited state (two bumps, antisymmetric). Do their energy values
differ, with respect to the values for the infinitely-deep square well, in the way
you expect? Do the wave functions look reasonable?
3. Find the energy values of all the bound states, and discuss how
they compare with the infinitely-deep square well.
4. Find the normalized wave functions for the second and the fifth
energy eigenvalues, and plot them together with the normalized wave
functions for the infinitely-deep square well. Give a brief discussion
of the comparison of the two. For the finite well, determine the
probability of the electron being found in the classically forbidden
region, for each of the bound state solutions.
Since you have left the height of the finite well as an adjustable parameter, and
since you know what the answers for the energy levels are for the infinitelydeep square well, you can check the functioning of your spreadsheet program by
increasing the (adjustable) height of your well to a big value (say 100,000 eV).
5. Discuss what happens when you do this what large value of the
well height did you use? What is the new ground state energy? Does
it agree with the trend you expected?
6.3
A Triangular Potential
Now that you have confidence with your spreadsheet method of solution, use it
to find the energy eigenvalues for the triangular potential:
V (x) = a|x| , with a = 2.6784462 × 10−6 J/m.
(6.14)
Technically this means there are two separate Schroödinger equations for the
regions of negative and positive x:
2
2
2
2
h
¯ d ψ
− 2m
dx2 − axψ = Eψ ,
for x < 0 , and
(6.15)
h
¯ d ψ
− 2m
dx2 + axψ = Eψ ,
for x > 0 .
We don’t have a solution for this potential in our textbook. Even if we did,
it wouldn’t be particularly enlightening because it involves special functions
6.3. A TRIANGULAR POTENTIAL
43
called “Airy functions”, that we would want to plot anyway in order to understand them. We can find them easily and automatically with your spreadsheet
solution.
6. Discuss what you expect the wave functions to look like, in both
wavelength and amplitude variation over the breadth of the potential.
Once again, due to the symmetry of the potential we expect either symmetric
or antisymmetric wave functions to result. So we can work with only the region
x > 0 as before, and either reflect or antireflect to get the solution for x < 0.
The latter of the equations 6.15 can be rewritten as:
E
2ma
x−
ψ .
(6.16)
ψxx = 2
a
¯h
Following the method used to solve the harmonic oscillator, and also in order to
simplify the spreadsheet calculation by scaling down the distance and energy,
introduce the variable change to get a dimensionless variable y:
y = βx ≡
2ma
h2
¯
1/3
x = 7.5988854 × 1010 m−1 x .
(6.17)
With this definition equation 6.16, transformed to an equation in y, becomes:
ε
ψyy = y −
ψ ,
(6.18)
b
where
b≡
a
=
β
¯ 2 a2
h
2m
1/3
= 3.52478827 × 10−17 J = 220.000000 eV
(6.19)
is the new energy scaling factor. This again allows you to use the energy parameter ε measured directly in eV. The reworked differential equation you’ll use
in your spreadsheet is then:
ε ψyy = y −
ψ .
(6.20)
220 eV
As with all bound state solutions (and this potential as specified has ONLY
bound state solutions), convergent solutions exist for only discrete energy eigenvalues, ε.
Modify your spreadsheet to find solutions for this potential. You can start
at y = 0 and use short steps of about ∆y = 0.1. You can figure out what upper
bound in y is appropriate. Determine the energy values of the first five bound
state energy levels and their corresponding wave functions.
7. Discuss the energies of the lowest five eigenfunctions in eV in
your writeup and comment on the spacing between them – is it even,
increasing, or decreasing? Is this what you might expect? Do the
wave functions look like what you predicted? Discuss.
44
Once you have the energy eigenvalues, you can adjust the non-zero starting
value of ψ(0) or ψy (0) that you used to generate the solution, in order to get
the integral of the probability density to be 0.5 for the range y > 0.
8. Include plots of the wave functions in your writeup for the first
five energy eigenfunctions (for y < 0 you can just reflect the wave
function across zero).
9. In your conclusion show the energies of the first five levels of all
three potentials, and discuss the variation of the spacings between
them. Also feel free to give any discussion about things you have
learned in this exercise about eigenfunction, quantization, etc.
Laboratory 7
Radioactivity
Equipment
SpecTech ST360 Interface Unit
Geiger-M¨
uller Tube with BNC cable
137
Cs/137 Ba Eluting Kit
Computer with ST360 software installed
Slotted holder station with tray
Various radioactive sources
Set of absorber plates
We are constantly and inescapably immersed in radioactivity from various
sources: electromagnetic radiation from the Sun, TV and radio stations, and cell
phones; cosmic rays (charged particles) from outer space; and even radioactive
sources that are built into our own bodies (about 1 in 10,000 of all potassium
atoms are radioactive 40 K, and roughly 1 in a trillion carbon atoms are radioactive 14 C). In this lab you will investigate the various types of radioactivity,
becoming familiar with their detection and also their penetrating properties.
There are three fundamental types of emissions from radioactive nuclei:
• Alpha radiation: the emission of a 4 He nucleus, a particulary tightly bound
combination of two protons and two neutrons, common in the highest-A
elements.
• Beta radiation: the emission of an electron (or positron) which changes
the Z value of the decaying nucleus by +1 (or 1) and is common in nuclei
with very small (or very large) values of Z/N , the ratio of number of
protons to number of neutrons.
• Gamma radiation: the emission of a high-energy photon directly from the
nucleus, usually as a by-product when a decay daughter nucleus is born
as an excited nuclear state. As a point of distinction, γ rays are emitted
from nuclei (and tend to be in the MeV range) and X-rays are produced
from inner-shell electrons (and tend to be in the keV range).
As you will see these radiations are quite different in nature.
45
46
LABORATORY 7. RADIOACTIVITY
IMPORTANT: Although all the radioactive sources in this lab are NRCexempted due to their relatively low activity level, please limit your exposure to them. Keep them in their plastic cases in the box of sources until
you are ready to use them. Be sure to put them away when you are done.
NO FOOD OR DRINK ALLOWED WHEN DOING THIS LAB!!
Familiarize yourself with the ST360 apparatus. Turn the unit on (the power
switch is on the back), and open the ST360 application by double clicking
on the icon which controls it (it may open automatically - if there isn’t an
icon on the desktop, look under: Programs → Spectech → Options → ST360).
This program controls the GM tube high voltage and records its count rates.
BE CAREFUL WITH THE GM TUBE. There is a thin mica window on the
end (to allow all radiation types including alpha to be detected) WHICH IS
VERY FRAGILE. If the GM tube isn’t mounted into the holder station already,
carefully remove its red protective plastic end cover before inserting into the top
of the measuring station, and connect the BNC cable for HV/readout to the
ST360 back panel plug. When you are done taking measurements, make sure
to turn the HV to zero (from the program menu), turn the ST360 unit off, and
make sure all sources are returned to the box. You can leave the GM tube in
the holder.
There are 7 parts to this lab, to be done over the course of TWO WEEKS:
1. Map the Geiger-Muller “plateau” to determine the tube’s optimum operating voltage.
2. Measure the background count rate.
3. Determine the GM tube’s “resolving time,” also known as “dead time.”
4. Measure the half-life of
137
Ba.
5. Qualitative familiarization with α radiation using the
210
Po source.
6. Study of the penetrating properties of β radiation using the
sources and absorber plates.
90
7. Study of the penetrating properties of γ radiation using the
and 137 Cs sources.
54
Sr or
Mn,
204
109
Tl
Cd
I suggest you do parts 1–4 the first week and parts 5–7 in the second week.
7.1
Finding the Optimum GM Tube Voltage
All GM tubes have their own optimum operating voltages, depending on the
size and geometry of the tube. The Tel-X-ometer unit has a smaller GM tube
which operates at about 425 V. The GM tube used in the ST360 is larger and
has a higher operating voltage. In this first part you’ll observe its operating
characteristics and assess what the best operating voltage is.
7.2. MEASURING THE BACKGROUND COUNT RATE
47
Place a 90 Sr source into the clear plastic holding tray (with the hole facing
up/foil label facing down) and install it into slot 2 of the measuring station.
Power up the ST360 and active its control panel (ST360 desktop icon). In
the Setup menu use the “HV Setting” option to control the high voltage. The
“Preset” menu allows you to take multiple runs (a series of count measurements)
for varying amounts of time. These are recorded into a data buffer you see in
the window, which you can write out as a file for use in EXCEL. Also notice
the button with a little “erase,” which clears the program’s data window.
Set the high voltage to 650 V and take a short run of 10 seconds to see
if there are any counts. Then gradually move up to 700 V, looking for where
counts begin to register. Once you have found the “threshold” take a series of
points from just below threshold voltage, up to 1100 V, measuring the count
rate at each one. The program will do this automatically for you:
1. Set the step option “On” in the “Step Voltage Enable” frame (“Setup”
menu).
2. In “HV Setting” set the step size to around 20 V, depending on your scan.
E.g., if you are starting from 660 V (first point with zero counts), you
can do 25 runs with a 20 V increment, which would take you from 660 V
through 1140 V.
3. In the “Preset” menu, set the time duration for each step to 30 seconds
(under “Preset Time”).
4. Set the number of runs to 25 (under “Runs”).
You should see a window for number of counts and data for all the runs and
the settings for high voltage and run parameters [If this window is not up, go
to the “View” option and select “Counts”]. Hit the “Erase” button to remove
any previous data, and then hit the green diamond button to start taking data.
Save a copy of the data, and then open EXCEL and read it in (read in the
“.TSV” file using the “IMPORT DATA” tab). You can now plot the count rate
as a function of voltage, and look for the “GM plateau.” You want a voltage
that is in the middle of a fairly stable count rate region.
What is the best operating voltage? Be sure to include a plot of this in your
report, AND INDICATE WHICH GM TUBE NUMBER YOU USED. One way
to assess the stability of the operating voltage in the central region of the plateau
is to measure the plateau’s slope. A “good” plateau can be considered to have
less than 10% variation in count rate (per second) per 100 V. Does your GM
tube have a “good” plateau? Will this value be the same for all the other tubes
in the lab? Will it be the same for this tube 10 years from now? Discuss.
7.2
Measuring the Background Count Rate
Cosmic rays and natural background radiation are an inescapable source of
background that you must quantify and then subtract from any radioactivity
48
measurements. Remove any sources from the area of the test station and set the
high voltage to the optimal operating voltage you determined in the previous
section (also turn off the “Step Voltage Enable” feature). Take twenty runs of
30 seconds duration, and combine these into an averaged background count rate
(per second) with error.
The distribution for a randomly-occurring count rate from a stable source,
such as that from this background or from a radioactive source, should a Poisson
distribution, which we discussed briefly last Fall:
PPoisson (n : µ) =
µn −µ
e
,
n!
(7.1)
where P (n : µ) gives the probability of observing n counts when expecting an
overall average rate equal to µ. Make a histogram of the count rates you observed
for the twenty 30-second intervals, and overplot the expectation from Poisson
statistics to see if they agree. I suggest you use bin intervals of 2 counts, to get
more statistics. Make sure to use 20 × P (n : µ) to match your sample size of
20 runs. You can overplot your measurements (points with error bars) with the
Poisson prediction (solid curve). Do they agree? Quantify the level of agreement
by giving a fit χ2 with NDOF.
7.3
Determining the GM Tube Resolving Time
When ionizing radiation traverses the interior of the GM tube, the strong electric field causes electrons and ionized gas particles to separate, rather than
quickly recombining as they would in a zero electric field. The liberated electrons accelerate, gaining enough energy to knock off more electrons, resulting in
an “avalanche effect” of a significant number of electrons reaching the central
cathode wire. This collection of charge registers as a count. The amount of
time it takes for this entire process to occur, which is also equal to the minimum amount of time necessary between tracks in order to be able to record a
second distinct count, is called the “resolving time,” or the “dead time.” If a
second track follows too soon after the first, the tube registers only one count,
of a longer pulse duration. You must estimate this resolving time, so you can
make corrections for it when the count rates are very high.
Let the true rate of ionizing tracks traversing the tube per second be R. Because of the resolving time, τ , the number of counts registered by the electronics
is a reduced value, R0 < R. Out of one second, the total dead time is R0 τ so the
counting efficiency will be (1 − R0 τ ). [It is assumed that the count rate is not
enormous, so there are very few “overlapping” dead hits.] Then the reduction
of count rate from its true value R to R0 arises from this efficiency factor:
R0 = R (1 − R0 τ ) .
(7.2)
The strategy for measuring τ is to use a pair of fairly intense calibrating
sources, and to observe the difference in the sum of the count rates obtained by
7.4. MEASURING THE HALF-LIFE OF
137
BA
49
measuring each source individually, compared to the rate measured when both
are present at the same time. Let the TRUE count rates be called R1 , R2 and
R12 , and suppose these are some fixed values. Then from equation 7.2 we have:
R10
= R1 (1 − R10 τ ) ,
R20
= R2 (1 − R20 τ ) , and
0
R12
0
= R12 (1 − R12
τ) .
By definition, a fourth equation also holds for the true rates:
R12 = R1 + R2 .
(7.3)
0
The above constitute a set of four equations in the four unknowns R10 , R20 , R12
and τ . They can be solved for τ , under the assumption that the difference
0
is much smaller than any individual
in measured count rates, R10 + R20 − R12
measured count rate. Show that when this assumption is made, you obtain the
following approximation for the resolving time:
τ=
0
R10 + R20 − R12
.
2R10 R20
(7.4)
The values inserted into this equation are the observed counting rates, not the
true ones. For example, if we recorded
R10
=
540 counts/sec,
R20
0
R12
=
640 counts/sec, and
=
1020 counts/sec,
then the inferred resolving time is 0.230 ms. This would imply complete saturation if the source were “hot enough” so as to approach 4000 or more tracks
per second impinging on the GM tube.
Among the source collection is a set of calibrating sources, in the shape of
half circles. Insert first one half-source into the tray, together with the blank
half circle (this helps keep the sources in exactly the same geometrical location
with respect to the end of the GM tube). Take 10 runs of data at 30 seconds per
run. Repeat for the second half-source in the location of the blank half circle,
and the blank half circle in place of the the first half-source. Then repeat for
the blank half-source replaced by the first half-source, so that both half sources
occupy the tray. MAKE SURE YOU REMOVE ANY SOURCES NOT BEING
MEASURED AWAY FROM THE AREA BEFORE TAKING DATA.
A reasonable resolving time is in the range of a couple hundred microseconds.
Be sure to state clearly your estimated resolving time and its uncertainty.
7.4
Measuring the Half-Life of
137
Ba
In this part you’ll observe the exponential nature of the activity for a shortlived source of radiation and compare it to an exponential decay curve, and
50
then extract its half-life from the data. The decay population of a sample of
N radioactive atoms is characterized by the lifetime, τ (not the same as the
resolving time!), which is the time constant in the exponential; and the closely
related half-life t1/2 which is the time taken for one half of the atoms to decay.
These are related by:
N (t) = N0 e
−t/τ
= N0 e
−λt
t/t1/2
1
= N0
2
(7.5)
where the decay constant, λ is related to the half-life by λ = ln 2/t1/2 . The rate
at which the population decreases is the count rate you observe in the detector
(when adjusted for angular acceptance). It is called the activity:
R=−
dN
= λN0 e−λt = λN .
dt
(7.6)
This is the law of radioactive decay – the activity, or decay rate is directly
proportional to the number of radioactive atoms preset. The rate decreases in
time with exactly the same exponential decay form as the number of radioactive
atoms left in the overall population.
The source we will use is the metastable isotope, 137 Ba∗ , which decays to
(stable) 137 Ba by gamma emission. This is an example of a nuclear energy
level transition which has a relatively long lifetime (“metastable”) because a
transition rule for nuclear energy level transitions is being violated. As you will
see the half-life is of the order of a few minutes. We obtain samples of it by a
clever process using elution which means “the removal of adsorbed material by
means of a solvent.” To generate a sample of 137 Ba∗ we use a canister of 137 Cs,
which is a longer-lived isotope (t1/2 = 30.1 years) that decays to barium-137
by beta emission. When radioactive cesium decays into barium its chemical
properties change from those of a Group 1 element to a Group 2 element. The
eluting solution (salt water with a bit of HCl) takes advantage of this to strip
out the freshly-formed 137 Ba∗ atoms, leaving the undecayed Cs-137 atoms of
the substrate behind. Most of the time (94.4%) the beta decay of cesium-137 is
to the metastable excited state of barium-137. Look up the details of this decay
online, and find out what the energy of the photon is that is emitted from the
decay of the metastable state.1 This is the energy of the photons your GM tube
will be detecting.
You should use 30 runs of 30 seconds each. Make sure the GM tube voltage
is at the optimum level you determined in Part 1. First measure the background
rate as you did in Part 2, taking 10 runs of 30 seconds. You can use the resolving
time from Part 3 for adjusting the number of counts. When you are ready to
start, obtain a small sample of 10 to 12 drops of Ba-137m isotope in solution
from Marcus or myself. [We will dispense this into a small aluminum planchet
for you – DO NOT do this on your own.] Immediately place the planchet with
1 One of the handiest reference sources is from the Korean Atomic Energy Research Institute, at: http://atom.kaeri.re.kr/.
7.5. QUALITATIVE PROPERTIES OF α RADIATION
51
solution into the plastic holding tray, mount it into the second shelf of the testing
station and start the run.
After the run has finished you can use EXCEL to analyze the data, and
from a plot of vs. time, you can get the decay constant. Apply corrections
to your data the “dead time” in the GM tube (using the correction factor you
found in Part 3, and equation 7.2), and for the backround rate. Once you have
completed this you can use linear regression with a plot of ln(count rate) to find
the half-life and its error. Make sure to lay out clearly how you do your data
analysis — the order of steps, the logic, the quality of the fit result, etc.
Is your measured value consistent with the standard value? Include a plot
of your data and the fit result, and your measured half-life, with error, in your
lab writeup.
7.5
Qualitative Properties of α Radiation
In this part you will carry out qualitative studies of the penetration of alpha
particles using a 210 Po source. Alpha particles are enormous in their effective
size, compared to electrons or photons. Additionally, the alpha will grab one or
two electrons as soon as it can (to become a helium atom) which helps stop its
progress even faster.
Check the background count rate by taking five 30-second runs of data with
the GM tube at its optimal voltage setting WITH NO SOURCE IN PLACE.
Then insert the 210 Po source into the top shelf, making sure that the foil/label
side is DOWN (the visible hole faces the GM tube). This is a short-lived isotope
(t1/2 = 138.4 days) so make sure you use one of the new sources (2008). This
shelf is situated only 1.5 cm from the GM tube front face. Record the count
rate for 30 seconds. Next, insert a single sheet of paper between the source and
the GM tube, and count the rate for another 30 seconds. Finally, remove the
paper and move the holder down to slot 2 and record for 30 seconds more.
Discuss the penetrating properties of alpha radiation. Does this mean sources
of alpha radiation pose no significant health risk? If not, then why is radon decay in basement areas considered a hazard? In 2006, Russian political refugee
Alexander Litvinenko died from poisoning by 210 Po – how was this possible,
considering the feeble penetration of α particles?
7.6
Beta Radiation
Beta particles are electrons emitted from the nucleus during decay, with energies
that are typically fractions of MeV. Due to their high energies, when such electrons enter matter they lose energy principally by Coulomb interactions with the
heavy nuclei of the atoms in the material. The process is called bremsstrahlung
(“braking”) radiation, and is discussed briefly on page 68 of your text. The
reduction in intensity of a beam of electrons entering matter is characterized by
52
the equation
I(x) = I0 e−x/X0 ,
(7.7)
where X0 is called the radiation length. For this part you will do a quantitative
study to estimate the radiation length of electrons in aluminum using a 90 Sr
source.
First, use explore the penetrating characteristics of β radiation in different
materials by placing the source in slot 2 and experimenting with absorbers of
different composition from the set. Is paper a good attenuator? Plastic? Foil?
Lead? How would you characterize what goes into making a good β radiation
absorber? Discuss this in your report.
Now leave the 90 Sr source in slot 2 and measure the count rates per second
for source alone, and then trying the various thicknesses of aluminum in the
absorber set. You can take 5 runs of 30 seconds for each of the steps. Plot
the logarithm of count rate as a function of thickness of aluminum in order to
measure X0 (include a plot with your report). You should find a value of about
0.10 cm. The attenuation length depends on the energy of the electrons. What
is the energy of the beta rays from your source? How does the attenuation
length change, as the energy is increased? Which process(es) of electron energy
loss dominates at lower energy, and which at high energy? Be sure to discuss
this in your report.
7.7
Gamma radiation
Photons interact, and lose energy when interacting with matter by the three
mechanisms we discussed in Chapter 2 of your text: the photoelectric effect,
Compton scattering, and pair production. The dominating effect(s) depend on
which energy range the photon is in. Taking as a scaling factor the electron
rest energy, me c2 = 0.511 MeV we can express the photon energy by x =
hν/me c2 . At lowest energies (x less than about 0.05) the photoelectric effect is
dominant. At highest energies, x > 2, pair production takes over. Most gamma
radiation falls between these thresholds, however and so Compton scattering is
the principle mode of interaction.
There are three gamma sources: 109 Cd, 137 Cs and 54 Mn. Look up the
energies of the gamma rays emitted by these sources. Then characterize the
general effectiveness of various absorbers on attenuating the radiation emitted
by these sources by using your absorber set. Does the energy of the gamma rays
have a significant effect on their penetration ability?
Finally, use the 54 Mn source to find the radiation length in lead, for the
energy of gamma rays emitted by this nucleus. You can proceed in the same
way as the radiation length determination in the previous section. A value
around 1 cm is expected here. How does your value compare? Again, include a
plot with the fit in your report.
Laboratory 8
Nuclear Spectroscopy
Equipment
Spectech NaI scintillator and photomultiplier tube
Set of 8 radioactive sources for gamma ray spectroscopy
Computer with UC30 software installed
The Geiger-Muller tube is a very useful device for detecting radiation. However it gives us little direct information about the characteristics of that radiation. In this experiment you’ll use a different type of detector that is well
suited for not only sensing the presence of gamma radiation, but also for giving
information about the energy of the gamma rays. This detector, illustrated in
Figure 8.1, comes as an integrated, sealed package comprising a NaI crystal encased in an aluminum tube, with a light connection to a photomultiplier tube.
The basic mechanism of detection is scintillation – the emission of light when
bombarded by radiation – which you already encountered with the phosphorescent rings in the e/m experiment. Here, the NaI (with a small admixture
of thallium) is the medium in which the scintillation occurs, under the stimulus of gamma radiation. The remarkable property of this process in NaI is
that the number of scintillation photons produced is proportional to the gamma
radiation energy.
The NaI crystal must remain sealed because it degrades under the influence
of humidity when exposed to air. The inside of the aluminum shield that covers it has a highly reflective material that causes nearly all photons that result
from scintillation to enter the photomultiplier tube, where the signals are significantly amplified. You should include a thorough explanation of the NaI
scintillator and photomultiplier tube in your report. You will need to
research the mechanisms of operation yourself (remember to include citations).
A manual for the UC30 unit will be made available to you, which includes at
least a good start in this direction. It also has useful reference material for
carrying out the investigations outlined below.
53
54
LABORATORY 8. NUCLEAR SPECTROSCOPY
high-voltage
cable
readout
cable
photomultiplier
tube
NaI crystal,
encased in Al
Pb shield
and mount
Figure 8.1: The integrated NaI crystal unit from SpecTech, which is operated
by the UCS30 computer interface.
Gamma rays that are emitted during nuclear decay do not change A, Z or N
but instead carry off energy from the transition of a daughter decay nucleus from
an excited state to a lower energy state or the ground state. As an example
of this consider the decay of 137 Cs, for which a decay diagram is shown in
Figure 8.2. This is a beta decay process and the daughter is 137 Ba. A small
fraction, 5.6% of the decays, result in the emission of an electron (and electron
antineutrino) and a daughter 137 Ba nucleus in the ground state. However, it is
more likely the case that the daughter barium nucleus is created in an excited
state – this happens the other 94.4% of the time. The excited-state barium
nucleus is “metastable” (has an unusually long lifetime). Its decay to the ground
state results in the emission of a 0.611 MeV gamma ray, which you detected
with a GM tube in your previous lab in order to measure the half-life of this
transition. This process is called an isomeric transition (IT). The NaI detector
senses the presence of 137 Ba∗ , and hence 137 Cs, by detecting this gamma ray.
There are numerous other decay processes resulting in the emission of gamma
rays of unique energy, and they constitute “fingerprints” for the presence of these
isotopes, similar to the way in which spectral lines can be used to identify which
elements are present in atomic spectroscopy. By providing energy information
the NaI crystal allows a similar identification process for certain nuclei, and this
technique is called nuclear spectroscopy.
Not all radioactive isotopes decays yield gamma rays – this requires an IT
or EC (electron capture) in their decay chain. Hence, only these isotopes can be
detected. In this laboratory you will examine some of them with this technique.
8.1. OPERATION AND CALIBRATION
55
Beta ray:
Max.E(keV)
1176( 1)
892.1( -)
514.03(23)
Gamma ray:
Energy(keV)
283.5( 1)
661.657( 3)
Avg.E(keV)
416.264(72)
300.570(68)
174.320(61)
Intensity(rel)
5.6( 2)
5.8E-4( 8)
94.4( 2)
Spin
7/2+
3/2+
1/2+
11/2-
Intensity(rel)
5.8E-4( 8)
85.1( 2)
Figure 8.2: Decay diagram for
137
Cs decay.
IMPORTANT: Although all the radioactive sources in this lab are NRCexempted due to their relatively low activity level, please limit your exposure to them. Keep them in their plastic cases in the box of sources until
you are ready to use them. Be sure to put them away when you are done.
Make sure to turn off the high voltage of the UC30 when you are finished.
NO FOOD OR DRINK ALLOWED WHEN DOING THIS LAB!!
8.1
Operation and Calibration
Turn on the power for the UC30 interface unit and then open the UC30 software
using the desktop icon. A display panel will open which allows you to control
the unit, calibrate it, take data, determine peak positions and write output files.
During measurements the photomultiplier amplifies light signals from the
NaI crystal, producing electrical pulses that are proportional to the energy of
the incident gamma ray. These electrical signals are converted into a digital signal by means of a 1024-channel ADC (analog-to-digital converter). The
amount of amplification that happens between an incident gamma ray and the
resulting pulse height (which is an ADC channel number between 0 and 1023)
depends on two parameters: the high voltage of the PMT and the gain of the
amplifying electronics that gather the PMT signal before conversion to a digital
value. These two parameters “set the energy scale” and it is important to verify/calibrate this conversion process before data can be taken. There are two
ways to calibrate:
1. Autocalibration: The UC30 has an autocalibration which uses signals from
a 137 Cs source. The autocalibration will systematically adjust both the
56
high voltage and gain setting in small steps (you can watch the progress
on your screen), moving these up until it sees a “healthy” peak for the
661.6 keV photon associated with this decay. It then calculates the scaling factor for the ADC and changes the digital scale along the bottom
of the display into an energy display. This method of calibration is useful for studying lower-energy gamma rays, since the calibration puts the
661.6 keV peak at the upper end of the ADC range. Gamma rays of
large energy (larger than 1 MeV or so) will be “off scale” and will not be
registered at all.
2. Calibration using known peak positions: You can use a reference source(s)
to tell the software how to set the ADC scaling. This method allows you
to set the PMT and gain factors to smaller values so that higher-energy
gamma rays can be measured. A suggested approach is to use as references
the 22 Na source (photons at 511 keV and 1274.5 keV) and the 54 Mn source
(photon at 834.8 keV), to perform a “3-peak” calibration. Use a high
voltage setting of about 775 V with coarse gain of 4 and fine gain of about
1.6, and take data so that you can see these three peaks clearly, and in the
range that you want them to be (you can put both sources at the same
time). After you have gathered some data, stop the run and set “ROI’s”
(regions of interest) at these locations. Select 3-peak calibration and enter
these three peaks and the known energies in the popup menus.
Whichever approach you take it is suggested that you periodically check the
calibration, especially if you start a new session on a different day, by using the
137
Cs source and verifying the peak location at 661.6 keV. If the position your
scaling gives differs by more than 10 keV you should recalibrate.
The software will determine peak position (“cetnroid”) and width (FWHM)
information, by setting ROI’s, accessed with a right click of the mouse. You
can print out a summary of peak information using the DISPLAY menu. If
you want to save a spectrum for later use with EXCEL or Mathematica, make
sure to save it in tab-separated-value format (extension names “.tsv”), which
is NOT the default given in the menu that appears when you use the SAVE
command.
IMPORTANT: When you are finished please be sure all sources are put
away, and that the UC30 high voltage has been turned off. You can do this
with the “SETTINGS” tab, where you see the HV ON/OFF radio buttons.
Then turn off the power on the interface box.
8.2
Energy Scale and Resolution
Complete a 3-point calibration as outlined above. For each one of the peaks you
identify in this part of the experiment note the measured peak energy, the ADC
channel numbers, and also the full width at half maximum (FWHM), which
you can obtain by setting a “region of interest” (ROI). Find the energies and
8.3. COMPTON SCATTERING
57
FWHM values for the 137 Cs and 60 Co sources. You should now have gamma
rays of six different energies.
Plot the energy vs. ADC channel number. Is it a purely linear relationship?
You might try a quadratic fit, which might do better in the case where there is
a small nonlinearity across the range of the ADC.
As you have noticed already, the width of the peaks increases with the energy. The number of photoelectrons from the PMT varies about a central value
in a purely “random walk” manner, which means the shape of a single gamma
ray peak is very nearly Gaussian. The width is expected to increase in a manner proportional to the square root of the energy. This relationship is usually
expressed in the form:
√
E
∆E
∝
,
(8.1)
E
E
where for ∆E we can use the FWHM. In order to linearize this, consider instead
the square of the quantity on the left, which should be fitted by a line:
2
∆E
1
=m
+b .
(8.2)
E
E
Make a plot of the quantity on the left, vs. 1/E, and see how well it fits a line
and also whether the intercept is consistent with zero, and discuss these in your
report. Improving the energy resolution (making m as small as possible) is an
important effort in scintillation detectors.
8.3
Compton Scattering
We can understand more about the structure of the typical gamma ray spectrum
by recall the mechanisms of Compton scattering, and X-ray production which
you studied earlier this semester. In Compton scattering an incident photon of
wavelength λ0 scatters from an electron of mass me (considered to be at rest)
and emerges with a different wavelength λ0 and with an angle of deflection θ
with respect to the original direction of travel. The change in wavelength is
given by the relationship for Compton scattering,
λ0 − λ0 =
h
(1 − cos θ) = λC (1 − cos θ) ,
me c
(8.3)
where λC = 2.426 pm is called the Compton wavelength. Looking at spectra you
have taken so far, it appears that most of the time a gamma ray enters the NaI
and undergoes a cascade whereby all its energy is converted into photoelectrons
detected by the ADC. However, if the gamma ray undergoes Compton scattering
then a different photon of energy E 0 given by:
1
1
1
−
=
(1 − cos θ)
E0
E0
me c2
(8.4)
is created, and a high-energy electron is created. When this occurs the photon
may exit undetected and the electron may be detected instead. There is evidence
58
for this occurring in your data, and the process is noteworthy because it forms
an inevitable background in many of the spectra you record.
The maximum energy electrons can acquire corresponds to a backscatter
event, θ = 180◦ . In this case the electron picks up energy equal to the difference
between the incoming and outgoing photons:
Emax (electron) =
2E02
,
2E0 + me c2
(8.5)
and the scattered photon achieves its minimum (“backscatter”) energy:
E0 me c2
Emin (photon) =
.
2E0 + me c2
(8.6)
An example spectrum for 137 Cs is shown in Figure 8.3. From the photon energy
X-ray
peaks
100000
Frequency
10000
Cs-137 spectrum
backscatter
“peak” @
185 keV
plateau
main peak
@ 662 keV
Compton
edge
1000
100
10
1
0
200
400
600
800
Energy (keV)
Figure 8.3: A spectrum plot from the 137 Cs source, with various features as
discussed in the text. Note that the frequency uses a log scale.
of 661.6 keV, we predict a “Compton edge” at 477 keV. This is the highest
energy that ejected electrons can have. In this spectrum only minor evidence of
a backscattered photon peak at 185 keV is seen. There are also X-rays peaks at
approximately 34 keV and 80 keV. The first is consistent with a Kα emission
from cesium or barium. These could be produced, of course, by scattered electrons liberating an inner shell electron in the material of the sample itself. The
second is consistent with a much heavier element – what do you suppose might
be causing it?
8.4. USING GAMMA SPECTROSCOPY
59
Use another gamma ray source, such as 54 Mn, and repeat this exercise of
identifying the Compton scattering features in your spectrum, and include a
plot in your report.
8.4
Using Gamma Spectroscopy
There is a source labeled “UNKNOWN” in the set of eight gamma ray sources.
Record a spectrum for this material and see if you can identify what’s present
there. The printed manual for the UC30 has useful reference material for this
in the appendices, especially the lists in Appendices E and F. Give your best
guess at what’s there, and see if you can say anything at all about the activities
of the unknowns in that sample.
Finally, if you wish, you may use the spectrometer to check for either environmental sources in the matter that’s in the lab area (tables, chairs, etc.) or
other samples of material you may wish to bring in. Scintillator detectors are
used principally in particle physics detectors, and also in forensics applications.
In the latter case, while most atoms of a given material are not radioactive, they
can be “activated” by flooding them with energetic neutrons at a reactor facility. The resulting sample will then contain neutron-heavy radioactive isotopes of
the elements comprising the material which usually have signature gamma rays
associated with their decays. This technique is called “neutron activation analysis,” and it can be applied successfully with very small samples of a material
of unknown composition (it is the subject of homework problem 12-42).
60
Appendix A
Review of Error Analysis
In this course an ”error” does not mean a mistake, or an admission of deficiency
in your measurement. Nor does it mean the difference between the measurement
of a quantity that you made and some ”established value” of the same quantity. ”Error” means your own best estimate of the precision of your
measurement. Although I refer to this quantity as ”error” it is completely
synonymous with ”uncertainty.” This section reviews some of the techniques for
estimating the error of your measurement, including the combination of error
contributions from multiple effects.
A.1
The error from the statistical error on the
mean
One of the most reliable ways to estimate statistical error is to repeat a measurement many times. Since you won’t get exactly the same answer every time you
make a measurement, you will probably get a cluster of measurements about
some central value. If these measurements are ”normally” distributed you’ll get
something that looks like a Gaussian ”bump.” The average, or mean value is
taken as your best estimate of the quantity you are measuring, and the error is
taken as the error on the mean, which equals the standard deviation divided by
the square root of the number of measurements:
σ
ε= √
N
(A.1)
As an example consider the distribution of ten measurements of the value of
electron’s charge-to-mass ratio shown by the histogram in Figure.
Note that this method requires a minimum number of measurements in
order to justify treating the distribution using normal statistics (this minimum
is quoted, variously, from 5 to 7). And note also this assumes your data are
61
62
APPENDIX A. REVIEW OF ERROR ANALYSIS
distributed normally1 . Always look at the distribution of your data in the form
of histogram! Are they distributed in a typical ”bump” fashion? Or are there
values that are a long way outside where the majority lie (called ”outliers”)? If
there are, find out what happened with these points! Remember, the procedure
outlined here will always give you values of the mean and the error on the mean –
it is up to you to decided whether it makes sense to do this with the distribution
of values you have.
Once you have determined the error, a convenient graphical comparison of
your result to a reference value can be made by plotting your measurement as a
point with error bars and the reference value as a straight line. Consistency at
the 1σ level is indicated by the error bar overlapping the reference line. For the
value of e/m for the electron from the above example, the comparison is shown
in Figure. This figure also allows an illustration of the difference between two
descriptions of a measurement that are often confused, or misunderstood.
• Precision refers to the size of the error on a measurement.
• Accuracy refers to the degree of consistency with a more established result,
or standard value of the quantity being measured.
In Figure you can see that a result can have either attribute without the other.
A result may be precise but inaccurate, due to, e.g., undiscovered systematic
error effects. It is also possible for a result to be accurate but imprecise due to
making too few measurements, or exaggerated estimations of error contributions
that cause the error bars to be larger than they should be. Ideally you should
achieve both precision and accuracy – error bars should, to the best of your
evaluations and estimations, truly represent 1σ confidence levels.
A.2
Error propagation
Occasionally you may need to find the error on a derived quantity, which depends
on one or more measurements you made and whose errors you have estimated.
For example, when using the Bragg scattering formula,
d sin θ = mλ
to estimating the lattice spacing, d, of a crystal one would measure the peak location θ to an error of εθ , that results from scattering of radiation of wavelength
λ corresponding to order of diffraction m. The measurement of d is obtained
from the measured peak angle θ, but what is the corresponding error on d?
In most cases this error is found simply by treating the input measurement
error(s) as differential inputs in a Taylor series expansion as a function of the
measured quantities. For example if we have a function f (x), then in the vicinity
of a point x0 the value of f is approximated by
f (x) ≈ f (x0 ) +
df
(x − x0 )
dx
1 The term ”normal” has precise meaning in terms of a Gaussian distribution, as discussed
in the second Appendix.
A.2. ERROR PROPAGATION
63
so that small excursions εx about value x0 translate into excursions of the function f of size εf about the central value f (x0 ) that are obtained from the scaling
factor df /dx evaluated at x0 :
df εf = εx .
dx
The absolute value is used because we are not concerned with the sign of the
derivative for plus-or-minus variations. This example shows how to treat the
error with respect to variation of a single dependent quantity. If there are several
dependent quantities, x, y, z, etc. the same idea holds in an extension of this
rule:
2
2
2
∂f
∂f
∂f
2
2
2
(εf ) =
(εx ) +
(εy ) +
(εz )2 + ...
(A.2)
∂x
∂y
∂z
Notice that errors are combined in quadrature – this is correct only if the errors
in the different measurements are uncorrelated, and allows for the fact that the
maximal deviation in one dependent variable does not generally occur in concert
with maximal deviation in another. Notice also that we have used only firstorder deviations (first derivatives) in these estimations. This approximation is
generally valid for cases where the error is much smaller in magnitude than the
quantity being measured (otherwise, it’s not much of a measurement, is it?).
Here are three special cases of the general relationship above that occur
frequently in laboratory analyses: our
1. Direct addition or subtraction of measured quantities results in a combined
error that is the quadrature sum of the participating measurements. For
example:
f (x, y, z) = ax + by + cz
has error found from:
2
2
2
(εf )2 = (aεx ) + (bεy ) + (cεz )
.
(A.3)
2. Direct multiplication or division of measured quantities results in a combined fractional error that is the quadrature sum of the fractional errors
of the participating measurements. For example:
f (x, y, z) =
xy
z
has error found from:
2 2 εf
εx 2
εy
εz 2
=
+
+
.
f
x
y
z
(A.4)
3. Power-law dependences multiply the error on the measured quantity by
the powers of the exponents. For example, if:
f (x, y) = xy 3
64
APPENDIX A. REVIEW OF ERROR ANALYSIS
then:
εf
f
2
=
ε 2
x
x
2
εy
.
+ 3
y
(A.5)
Therefore, large powers exacerbate the dependence of the overall error on
that particular measurement. In this case, the relative precision of the
measurement of y has three times the effect of the relative precision of
the measurement of x in determining the overall precision of the derived
quantity.
In the example of the Bragg scattering formula above we have a functional
relation that is not directly treated as one of these special cases. If the interplanar separation distance d is considered a function only of θ,
d=
then
mλ
sin θ
dd mλ cos θ
εd = εθ =
= d cot θ .
dθ
sin2 θ
The last form is especially revealing – since the same distance d results from
measurement at any order of diffraction, it says that if angular uncertainties are
the same at all angles of measurement then greater precision on d is obtained
from the (higher order) diffraction angles closest to 90 degrees.
A.3
Combining several measurements of the same
quantity
Several measurements of the same quantity, xi , i = 1, ...N , each with error εi
can be combined into a single measurement:
PN
PN
xi /ε2i
i=1 wi xi
=
hxi = Pi=1
PN
N
2
i=1 1/εi
i=1 wi
(A.6)
provided that none of the errors are correlated. The inverse-error-squared is often referred to as the weight of the measurement, since a relatively smaller error
results in more ”power” in determining the final overall value. The combined
error is then:
v
v
u
uN
N
u X
uX
t
2
1/εi = t
wi .
(A.7)
εx = 1/
i=1
I=1
The overall error is always smaller than the smallest of the individual measurement errors – you can only improve by adding more measurements.

Modern Physics II Lab Manual University of Puget Sound Spring Semester 2014

Transcription

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