Experiment 1. STATISTICS IN COUNTING EXPERIMENTS
From: www.wsu.edu/~collins/Phys415/writeups/gammaabs.htm
Emissions from a long-lived radioactive source are thought to occur randomly in time.
The distribution of time intervals between these emissions is then described by the
Poisson distribution. In this experiment, you will gain a thorough understanding of the
statistics in counting experiments in which events occur randomly in time. This
knowledge is basic to interpretation of many of the experiments you will carry out later
in this course.
Equipment: Geiger detector, scintillation detector, time-interval measuremenr (period
measurement and scaling)..
Readings: Radiation and detectors (general), sect. 5.1; Interaction of charged particles
with matter, 5.2.2; Geiger detectors, sect. 5.3.1 and 5.3.4; Counting statistics, sect. 10.5
and and 5.3.5. [Unless noted otherwise, all readings are from A. Melissinos,
Experiments in Modern Physics,(Academic Press, New York 1966)]; Data Analysis for
Physical Scientists, Louis Lyons (Cambridge, 1991), chapter 1.
Key concepts: Random processes, empirical frequency distribution functions, Poisson
and Gaussian distributions; Geiger tube operation (plateau, continuous discharge region,
operating point, dead time).
1.1 Counts detected in a fixed time interval
Instrumentation here includes a scintillation detector (Ludlum) and frequency counter.
The scintillation detector generates an output pulse for each pulse that is detected above
a certain threshold voltage, with no information about the energy of the pulse. The
active element of the counter is a 1-inch by 1-inch NaI crystal. The frequency counter
acts as a pulse counter. For each of parts 1 and 2 below, measure in 100 trials the
number of radiations detected in a given time interval by the scintillation counter. You
may either count cosmic and background radiation without a radioacive source nearby
or also radiation from a beta source placed near the counter.
1.1.1 Small average number of counts
Select a time interval so that the mean number of counts is very close to 2, using lead
shielding if necessary. Make a frequency histogram for 100 trials, and compare your
normalized histogram against the hypothesis that the cosmic-ray flux is timeindependent, on average, with the same average counting rate, using Poisson statistics.
Compare the standard deviations of experimental and theoretical distributions having
the same mean.
1.1.2 Large average number of counts
Select a time interval and arrange shielding so that the averate number of counts is close
to 100. Make a frequency histogram for 100 trials and compare your normalized
distribution to the Gaussian approximation of the Poisson distribution having the same
mean counting rate. This approximation is excellent when the mean number of counts
in the interval is large (greater than about 5-10). Compare standard deviations of
experimental and theoretical distributions having the same means.
1.2 Distributions of time intervals between counts
Instrumentation here includes a Geiger detector and time interval counter run in 'period'
mode. The Geiger detector has an active volume of about 30 cm3 and is most sensitive
to impinging charged particles. About 1 cosmic ray (mostly muons) will be detected
per second for that volume. In period mode, a counter starts a clock when a first count
is detected and counts periods of an oscillator until a second pulse is detected. The
displayed count is a measure of the time interval between successive pulses.
1.2.1 Between successive counts
Make 100 measurements of time intervals between successive pulses using the Geiger
detector. Histogram the distribution of time intervals using an appropriate bin-width,
and compare your distribution with the prediction for a random series of pulses that are
Poisson distributed and having the same mean counting rate.
1.2.2 Between every other count
A divide-by-n circuit may be available that can be connected to the output of the Geiger
counter. Depending on the switch setting, only the 2nd, 4th or 8th pulse following an
initial pulse will generate an output. One can thus measure the distribution of timeintervals between a pulse and the 2nd, 4th or 8th subsequent pulse from the circuit
output. Make 100 measurements as in part 1.2.1 for time intervals to the 2nd pulse.
Compare experimental and theoretical distribution functions for the same average
counting rate.
If you have time and interest, you may do the same for the distributions of intervals to
the 4th or 8th pulses. In the absence of a divide-by-n circuit, you may instead sum pairs
of intervals between subsequent pulses to get a representation of the distribution of time
intervals until a second pulse is detected. Compare experimental and theoretical
frequency histograms.
Questions and Considerations
a. Know how a Geiger detector works. If it is possible to vary the operating voltage of
your tector, determine a safe operating point by measuring the counting rate as a
function of the applied high voltage. Never allow the tube to operate in the continuous
discharge region, which will destroy the tube!
b. For what kinds of radiation is the Geiger detector most sensitive? Why?
c. Can a Geiger detector be used to measure the energy of incident radiation? Explain.
d. The Geiger tube has a dead time of ~300 microseconds following each pulse. Can
you think of a way to measure the dead time? Do you need to take the dead time into
account in or any of the measurements in this experiment?
Diagram of Apparatus
Copyright Gary S. Collins 1997-2002..
Experiment 4. GAMMA-RAY ABSORPTION
In this experiment you will measure the transmission of gamma rays through different
absorbers. Theoretically, there should be an exponential decrease of the transmitted
counts with thickness of absorber, determined by a mass absorption coefficient. You
will determine mass absorption coefficients and evaluate how they vary as a function of
gamma energy and of the atomic number of the absorber.
Equipment: NaI scintillation detector, amplifier, pulse-height analyzer; gamma sources,
sets of absorber plates.
Readings: Interactions of photons with matter, concept of crosssections, sect. 5.2.1 and
5.2.5; Absorption measurements, sect. 5.4.3.
Key concepts: Absorption and scattering crosssections for photons; absorption lengths,
mass absorption coefficients.
4.1 Transmitted counts vs. absorber thickness
Absorbers of Al, Cu, Cd and Pb are available in plates that can be stacked to produce a
range of thicknesses. Various gamma sources are available, including 137Cs (662 keV),
60
Co (1.17 and 1.33 MeV) , 57Co (122 keV), 22Na (511 keV, 1.27 MeV) , and 241Am
(59.7 keV) may be available. Also, some sources emit x-rays of lower energy, e.g. K xrays of Ba followind decay of 137Cs. For various paired choices of source and absorber,
make measurements of transmitted counts as a function of absorber thickness. Choose a
range of absorber thicknesses that result in a large change in the number of transmitted
counts and make a series of measurements to help experimentally to determine how the
number of transmitted counts decreases with absorber thickness . Measure absorber
thicknesses with a micrometer or vernier caliper.
An important issue is to determine the "background" counting rate that is irrelevant to
your measurements. Background radiation may come from cosmic rays and
environmental radioactivity, but also, e.g., from scattering of radiation from your source
off of nearby objects into the detector. You need to use judgement to determine best the
background counting rate. After subtraction of background counting rates, plot count
rates on semi-log paper versus absorber thickness. Determine absorption lengths and
coefficients from slopes using the theory of gamma ray absorption (exponential
absorption law).
Compare your measured coefficients with those obtained from the attached graph of
mass absorption coefficients or from the net (e.g. see http://www.csrri.iit.edu/periodictable.html or other links on the course home page). Ascertain whether or not your
coefficients are consistent with those reported elsewhere. If they are not consistent, try
to figure out why and to explain how better values might be obtained.
4.2 Mass absorption coefficient versus gamma energy
Using one absorber determine absorption coefficients for a wide range of gamma
energies. Al, Cu or Cd is recommended over Pb. Plot coefficients versus energy and
try to establish a qualitative or quantitative dependence,
4.3 Mass absorption coefficients vs. absorber atomic number Z
Using one gamma source (preferably 57Co or 241Am) determine absorption coefficients
for absorbers having a wide range of atomic numbers Z. Plot mass absorption
coefficients versus Z and try to establish a qualitative or quantitative dependence.
Questions and Considerations
a. How can you determine the "true" background counting rate? By interposing a very
thick absorber? By moving the source far away? Are there contributions to
"background" from the source itself? If so, could they be reduced through a better
experimental design?
b. The total crosssection for removal of photons from the beam, or 'extinction'
crosssection, is the sum of crosssections for photoelectric absorption, Compton
scattering and pair production. How might you try to distinguish between these
contributions experimentally?
c. Consider the simple pulse-height spectrum of 137Cs, which emits a single gamma ray
with an energy of 662 keV. To determine the counting rate, you have the choice of
integrating counts over the entire spectrum or only over the photopeak. What are the
tradeoffs in each choice?
Diagram of Apparatus
Mass absorption coefficients for selected elements
Energy units BeV= Billion of electron-volts= GeV. Note that lower curves are
extensions of the upper curves to higher energies and that there are different energy
scales at top and bottom of the graph..
.
Copyright Gary S. Collins, 1997-2002..
Experiment 5. X-RAY ENERGIES OF THE
ELEMENTS (MOSELEY'S LAW)
Photoelectric absorption of high-energy photons in an atom is accompanied by ejection
of an inner electron from the atom. The excited atom then returns to its ground state
with emission of x-rays or Auger electrons. K x-rays are emitted when an outer electron
fills a "hole" left in the K shell. In 1913, Moseley published measurements of
characteristic energies of K x-rays of many elements, explaining them in terms of the
then-new Bohr atomic theory. Moseley showed that the energies were given in good
approximation by:
EK = 3/4 (Z - b)2 EI,
in which Z is the atomic number of the element, b is an empirical screening constant
roughly equal to , and EI is the ionization energy of the hydrogen atom, 13.6 eV. The
findings lent great support to Bohr theory, which of course was proposed as an
explanation of optical spectra. In this experiment you will excite x-rays from various
elements by placing samples in a gamma-ray beam from a radioactive source. The xrays will be detected using a gas proportional counter.
Equipment: Gas proportional counter, pulse amplifier, pulse-height analyzer. Samples
of elements and compounds. Gamma-ray source such as 241 Am or 57Co to excite x-rays.
Readings: Proportional counter, sect. 5.3 (especially 5.3.3). Know how to derive
Moseley's law given by the equation above (consult an introductory modern physics
text). Become familiar with the decay scheme and radiations of the gamma source you
will use to excite x-rays (57Co (page 272) or241Am).
Key Concepts: Production of x-rays by photoelectric absorption. K and L x-ray
energies. Escape peak.
5.1 Energy calibration.
Keep counting rates low during calibration to avoid possible shifts when counting rates
are high. Place sources some idstance from the proportional counter, with the counter
shielded except for the (delicate) beryllium entrance window.
For 57Co, three peaks should be present in the pulse-height spectrum: the 6.4 keV Kxray of the daughter nucleus 57Fe, the 14.4 keV gamma ray of 57Fe, and a Rh or Pd xray, depending on the matrix in which the 57Co activity was diffused. For 241Am,
photopeaks will be observed for 1 or 2 gamma-rays and for a bunch of Np daughter xrays. Gamma-rays are at 59.54 keV and (harder to see) 26.3 keV. X-rays from the Np
daughter are at 11.87, 13.93, 15.86, 17.61, and 21.00 keV. Other sources can also be
used to provide calibration peaks.
Use these energies to calibrate the PHA and then adjust amplifier gain so that energies
up to, say, 100 keV can be stored in the PHA.
5.2 Characteristic K x-ray energies
The higher energy 59.5 or 122 keV gammas which follow decay of 241Am or 57Co,
respectively, are absorbed in the target, followed to emission of characteristic x-rays.
Both K or L x-rays will be produced, but the efficiency of the detector decreases at low
and high energy so that both x-rays may not be detectable for a specified element
(why?).
Direct an intense beam of gammas from the source onto a sample, but use a collimator
and geometry as shown in the diagram to ensure that neither the beam nor scattering
from objects other than the sample is detected. The x-ray flux from the sample will be
low, so efforts to maximize the solid angle between target and counter window wil
avoiding any direct beam from the gamma source will pay off in improved statistics.
(see diagram below).
Measure characteristic x-ray energies for as many elements as you can. To linearize the
data, plot the square root of EK vs. Z for known and unknown elements, and identify the
unknowns. Indicate experimental uncertainties on your plot. From the slope and
intercept determine the ionization energy of hydrogen, the screening constant b.
Estemate their uncertainties.
5.3 L x-ray energies
K x-rays of high Z elements like W and Pb may not be observable (why?) and only L xrays may be detected. Read about Moseley's extension of his theory to explain L x-ray
energies, and test it experimentally.
Questions and comments
a. What advantages does a gas proportional counter have over Geiger or scintillation
counters for measurement of photon energies in the range 10-100 keV?
b. The method of x-ray production used here is nondestructive. Could the method be
used for quantitative analysis of elemental composition? How? You might test this by
analysis of the content of an alloy. For example, the nickle coin is reported to have a
content of about 75% Cu and 25% Ni.
c. Different x-rays may be produced at the same time in a sample. Qualitatively, what
determines the relative number of K and L xrays? How does the relative number
depend on the energy of the gamma ray and on properties of the element?
d. If you used a 57Co source, you will observe that the Pd or Rh x-ray near 21 keV is
much less intense than the Fe x-ray, although there are in fact very few Fe atoms in the
source. Explain.
e. Nuclear decay via electron capture (e.g. in 57Co) leaves the atom in an excited state
with, usually, one missing K or L-electron. Will deexcitation of the atom always lead to
emission of an x-ray? If not, what are alternative processes? Beta-emission, on the
other hand, is not obviously followed by x-ray emission. However, the pulse-height
spectrum of 137Cs, for example, exhibits a large photopeak for Ba K-xrays. Why?
f. In pulse-height spectra, escape peaks are attributed to escape from the detector
without detection of x-rays created during gamma absorption. X-ray escape is a more
probable process in a gas counter than in a solid counter of similar size owing to its
lower density. Most general purpose proportional counters are filled with Kr or Xe gas
(typically plus a small amount of CO2 to quench the discharge). Under what conditions
should escape peaks be more likely to be observed? Explain quantitatively if you can.
g. One might alternatively excite characteristic x-rays using bremmstrahlung radiation
from an x-ray tube. Very high fluences of exciting radiation can be generated..
Characteristic xrays can be detected with better energy resolution them using a
semiconductor detector.
Copyright 1997-2002 by Gary S. Collins.
Experiment 3. BETA-RAY ABSORPTION
You will measure the transmission of beta rays through thin foil absorbers of
aluminum.to determine the range of the betas. Since a beta energy spectrum is
continuous, the range should be related to the maximum kinetic energy of the betas. For
accurate range measurements, you will need to carefully subtract "background"
counting rates.
Equipment: Geiger detector, beta sources, set of foils, simple platform to support foils,
scaler.
Readings: Interactions of charged particles, 5.2.2 and 5.2.6; Geiger detectors, sect.
5.3.1; Range measurements, sect. 5.4.3.
Key concepts: Energy loss of a charged particle, and its range; range versus energy;
different sources of background counting rates.
3.1 Beta absorption measurements in Al
Sources are available that emit essentially only beta rays (e.g. see table below).
Choosing at least three sources with a range of maximum beta energies, make
measurements of the numbers of betas transmitted through stacks of Al foils of varying
thickness and detected in the Geiger counter. You will need ~100 small sheets of
standard kitchen Al foil (e.g., "Reynold's Warp"), which are usually quite close to
~0.001" thick. Determine the counting rate as a function of absorber thickness. The
challenge in this experiment is to determine the thickness for which all beta-rays are
stopped. Obtain statistically significant measurements of rates for thicknesses in the
range where the "true" transmitted counting rate has dropped to about 1/10 of the
background rate. Note that this may imply lengthy mesurements. By "background" is
indicated a counting rate that remains constant for thick absorbers over the range of foil
thicknesses studied. After subtracting "background" from your raw data, plot the
transmitted counts versus Al thickness on semi-log paper (as illustrated in fig. 5.32 in
Melissinos). Indicate the background level on your plot. Draw error bars for each
datum that reflect random counting errors and propagated errors from background
subtraction. Draw a smooth curve through the data "to guide the eye" and make your
best estimate of the range of betas for each source (with experimental uncertainty) in
units of mg/cm2 of Al. The table gives the maximum energies of betas for sources that
may be available.
Beta sources
Max. kinetic energy
(MeV)
14
0.155
C
99
Tc
0.29
Cl
0.715
36
204
Tl
0.763
210
Bi
1.17
234
Pa
2.32
3.2 Beta range in other absorbers
Make range measurements using sheets or thicknesses of another absorber such as
plastic film. Determine the range in mg/cm2 as in part 3.1 above. Compare the ranges in
Al and the other absorber. Try to explain any differences using a simple model to
describe the energy loss of electrions in the different materials. .
3.3 Range-energy relation
Correct your measured ranges for the energy-loss of betas in the air between source and
detector and in the thin mica window of the Geiger counter (thickness ~2.0 mg/cm2).
Plot corrected ranges versus maximum beta energy. Compare your values with
Feather's formula (eq. 5.2.13) or with the range-energy plot given below. .
3.4 Crude determination of the beta spectrum
Assuming that the energy-loss per unit thickness of absorber is independent of energy,
the derivative of the transmitted counting rate with respect to absorber thickness should
yield the energy spectrum of the source. Make a plot of the beta spectrum by taking the
derivative of your background corrected counting rates. While the result is more
qualitative than quantitative, you should be able to demonstrate the broad, continuous
nature of the beta-spectrum. (More precise measurements of the energy spectrum can
be obtained using a magnetic spectrometer or semiconductor detector such as the highpurity Ge detector used in th high resolution spectroscopy lab.)
Questions and Considerations
a. Enumerate possible sources of "background" counting rates. Consider different types
of radiation and their penetrating ability. Also consider if any background might be
attributable to the source. What empirical method might you use to determine the
background? Could Bremsstrahlung play a role?
b. Consider how best to plan your measurement times in order to obtain the most
precise values of the background-corrected counting rates.
c. Beta ranges, when expressed in units of mg/cm2, are experimentally found to be
approximately the same for different absorbers. What is the theoretical justification for
this trend?
d. 137Cs undergoes beta decay to an excited state of 137Ba, which in turn decays to the
ground state of 137Ba with emission of a 0.662 MeV gamma ray. Is it possible to
determine the range of betas emitted from 137Cs by the method used in this experiment?
Give an argument or (better) try it.
Diagram of Apparatus
Range of beta particles as a function of initial beta energy.
Copyright Gary S. Collins, 1997-2002.