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					Physics 340                                                                     Spring (3) 2010

                                   Nuclear Physics Lab II:
                    Geiger-Müller Counter and Nuclear Counting Statistics

PART I Geiger Tube: Optimal Operating Voltage and Resolving Time
Objective: To become acquainted with the operation and characteristics of the Geiger-Müller (GM)
counter. To determine the best operating voltage and the resolving time of a Geiger counter. The
resolving or dead time is used to correct for coincidence losses in the counter.

Experimental Apparatus: A typical Geiger-Müller counter consists of a cylindrical gas filled tube,
a high voltage supply, a counter and timer. A large potential difference is applied between the tube
body which acts as a cathode (negative potential) and a wire down the tube axis which acts as an
anode (positive potential). The sensitivity of the instrument is such that any particle capable of
ionizing a single gas molecule in the GM tube (thus producing an electron-ion pair) will initiate a
discharge in the tube.
What happens next depends on the voltage across the gas-filled tube. For the lowest applied
voltages, only the ions created by direct interaction with the incoming radiation are collected. In this
mode, the detector is called an ion chamber. For higher voltages, the ions created are accelerated by
the potential difference gaining sufficient energy to create more ion pairs. This results in a localized
avalanche of ions reaching the wire. This is the proportional region. The pulse height (or voltage of
the signal) is proportional to the number of initial ion pairs created by the incoming radiation. This
in turn is proportional to the energy of the incoming radiation. For even higher voltages, the new
ions can create additional photons which move out of the local region and further down the tube;
essentially the discharge propagates an avalanche of ionization throughout the entire tube, which
results in a voltage pulse--typically a volt in amplitude. Since the discharge is an avalanche and not
a pulse proportional to the energy deposited, the output pulse amplitude is independent of the energy
of the initiating particle and, therefore, gives no information as to the nature of the particle. This is
the Geiger-Müller region. In spite of the fact that the GM counter is not a proportional device, it is
an extremely versatile instrument in that it may be used for counting alpha particles, beta particles,
and gamma rays. Such a large output signal obviates the need for more than a single stage of
amplification in the associated electronic counter.
Geiger-Mueller tubes exhibit dead time effects due to the recombination time of the internal gas ions
after the occurrence of an ionizing event. The actual dead time depends on several factors including
the active volume and shape of the detector and can range from a few microseconds for miniature
tubes, to over 1000 microseconds for large volume devices. When making absolute measurements it
is important to compensate for dead time losses at higher counting rates.

Please keep all sources in the lead brick house. Take out only the one you need, and return
                     it as soon as you are done taking a measurement.

Optimal Operating Voltage: Procedure
[This section requires the GM tube from the Tel-X-Ometer X-Ray apparatus. If another group is

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Physics 340                                                                    Spring (3) 2010

currently using this detector you will need to negotiate with them for its use. You only need it for 30
minutes or less and you can do this part of the experiment at any time (so go on to the next part if the
X-ray GM tube is not currently available).]
Using the Geiger tube from the Tel-X-Ometer, with the voltage set at about 400 V, find the most
active source among those in the lead brick house. Simply set the Geiger counter on the table and
balance a source right up against the end of it. You will need to determine which side of the source
is the most active. Once you’ve identified the most active source, use that to measure voltage versus
counting rate. Starting at zero volts, take readings up to about 490 V. The counting rates will vary
significantly, so watch the number of counts for at least 30 seconds and record what seems to be the
average. In particular, you will want to take careful reading around the "turn-on" voltage. Estimate
an uncertainty for each reading.

Graph the voltage vs. counting rate. On your graph, identify the regions from those described in the
Introduction. Are all regions present? If not, why do you think they might not be represented?
Given your data, identify the optimum operating voltage range for this GM detector.

Dead Time: Theory and Procedure
As noted above, once a discharge has been initiated in the GM tube a new pulse will not be detected
until the previous discharge has extinguished itself. Thus there is a dead time τ associated with each
counting event. If we measure a counting rate r in a time interval Δt, the total detector dead time
will be Tdead = (rΔt)τ. Thus the true counting rate R and the measured counting rate r are related by
R(Δt – Tdead) = r Δt so if we know τ we can correct our measured count rates as follows
                                        r
                                R=          .                                                  (1)
                                     1 – rτ
We can determine τ by measuring the individual and combined counting rates from two high flux
samples. If we call these rate r1, r2, and rc we have the following three equations:
                    €
                               r1 = R1(1 – r1τ)                                      (2a)
                               r2 = R2(1 – r2τ)                                      (2b)
                               rc = (R1+R2)(1 – rcτ)                                 (2c)
with the three unknowns R1, R2, and τ. Solve this set of equations for τ and show that the dead time
is approximately given by:
                                  r +r −r
                             τ= 1 2 c .                                                      (3)
                                     2r1r2
Here we will make such measurements using our 137Cs sources and DataStudio to determine the dead
time for the PASCO Geiger counter. (Note that the operating voltage for this detector cannot be
adjusted). Set up the€DataStudio for a 30s counting interval. Devise a way to place the 137Cs sources
so that when both are in place they touch one another, are positioned midway between the ends of
the tube, and so that each source can be removed then replaced in exactly the same position.

With only one cesium radioactive source in place, take a five-minute count (10 measurements). The
count rate should be in the range 10,000-20,000. Record the count as r1. Place the second source

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Physics 340                                                                            Spring (3) 2010

beside the first (being careful not to disturb the first) and take a five-minute count of the combined
sources. Record this count as rc. Now remove the first source and take a third five-minute count.
Record this as r2. Repeat with the source positions reversed–because these sources are not of equal
strength. (Note: If the count rate exceeds 65,000 DataStudio will "reset" its counter).

Calculate the dead time τ of the PASCO GM detector for both arrangements using Eq. (3).

Now that we know τ for the PASCO GM tube we can use Eq. (1) to correct any counting rates
measured with this detector. Apply such a correction (if necessary) for data taken in the next
section.

PART II Statistics of Nuclear Counting*
*[Portions of the Theoretical Background are taken from Experimental γ -Ray Spectroscopy and
Investigations of Environmental Radioactivity Experiment 9 by Randolph S. Peterson, Spectrum
Techniques, Inc.]
                                                              €
Objective: To study the statistical fluctuations which occur in the disintegration rate of an
essentially constant radioactive source (one whose half-life is very long compared to the time
duration of the experiment).

Theoretical Background: We can never know the true value of something through measurement. If
we make a large number of measurements under (nearly) identical conditions, then we believe this
sample’s average to be near the true value. Sometimes the underlying statistics of the randomness in
the measurements allows us to express how far our sample average is likely to be from the real
value. Such is the situation with radioactive decay, with its probability for decay, λ, that is the same
for identical atoms.
Radioactive materials disintegrate in a completely random manner. There exists for any radioactive
substance a certain probability that any particular nucleus will emit radiation within a given time
interval. This probability is the same for all nuclei of the same type and is characteristic of that type
of nucleus. There is no way to predict the time at which an individual nucleus will decay. However,
when a large number of disintegrations take place, there is a definite average decay rate which is
characteristic of the particular nuclear type. Measurements of the decay rate taken over small time
intervals will yield values which fluctuate randomly about the average value and consequently which
follow the laws of statistics. Hence in dealing with data from measurements of radioactivity, the
results of the laws of statistics must be applied.
Given that λt is the probability of decay for a single nucleus in time interval t (and thus 1–λt is the
probability for non-decay), the probability P(n,t) of n nuclei decaying in time t from a sample of N
identical atoms is given exactly by the binomial distribution
                                                       N!
                                        P(n,t) =              ( λt) n (1− λt) N −n .                 (4)
                                                   n!(N − n)!
The mean and variance of this distribution are µ = Np and σ2 = Np(1–p), respectively, where p = λt.

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    Physics 340                                                                                         Spring (3) 2010

    If λt is small and N is large such that µ = λtN remains small, this binomial distribution can be
    approximated by the Poisson distribution
                                                   µn
                                          P(n,t) = e− µ                                      (5)
                                                   n!
    where µ = λNt is the average number of decays in time interval t.
                                  such
    If λt is small and N is large € that µ = λtN is not small (perhaps greater than 100), the binomial
    distribution can be approximated by the normal (Gaussian) distribution function,
€                                                                    −( n− µ )2
                                                         1
                                                                                           .                          (6)
                                                                                       2
                                          P(n,t) =               e                2σ
                                                             2
                                                     2πσ
    where σ 2 ≈ µ is the square of the standard deviation, and gives a measure of the width of the
    distribution. Experimentally we can approximate µ with our sample average A , and the standard
    deviation σ with the square root of A . Given M measurements of a source's activity, A, the
                               €
    frequency ƒ(A) with which we measure A (per interval ΔA) is
€
                                           €        MΔA −(A −A )
                                                                                           2
                                                                                                    €
    €                           € f (A) = MP(A)ΔA =      e                                     2A
                                                                                                                      (7)
                                                    2π A
    Note that ƒ(A) is the number of times our measurement falls in the range A → A+ΔA.

                        €




           FIGURE 1. Gaussian fit to counting frequency (using a 10 cps bin width) for 60Co.

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Physics 340                                                                     Spring (3) 2010

The data in Fig. 1 are compiled from 1,024 consecutive measurements of the number of detected
gammas per second emitted by a 60Co source. The frequency of the measured counting rates is well
represented by the Gaussian distribution curve of Eq (7) with A =7,540. Only a calculation of the
sample average and the total number of samples were necessary to calculate the distribution curve.
Eq. (4) could also be used to model the data of Fig. 1, however, application of Eq. (4) can be difficult
when dealing with large numbers. To see this, try calculating values for the frequency using Eq. (4)
                                                        €
for the 1,024 count rate measurements represented in Fig. 1.

In an actual experiment, there is always some background radiation present. This background is
mostly due to cosmic radiation reaching the earth, but is also composed of radiation from very small
amounts of radioactive material present in the walls, floor, and tables of the laboratory. If the
intensity of radiation from the radioactive material being used is very large compared to the
background, then the background may be ignored. If, however, a weak radiation source is being
used, it is important to subtract the background in order to determine the decay rate of the
radioactive material itself. In this experiment, correction for the background will not be necessary.
However, the background radiation will be used as a very low intensity source of radiation, hence
the Poisson distribution will best approximate the data.

In this experiment on statistics of nuclear counting, the rate at which radiation reaches a detector
from a long lived radioactive source is determined by measuring the number of events occurring in
the detector during a time interval of one minute. Many measurements are made and the average is
calculated from the values obtained. Even though each value represents the measurement of the
same quantity, the values will be different. The cause for the differences is the statistical fluctuation
in the amount of radiation reaching the detector. A careful study is then made of the fluctuations,
and the precision of the measurement is determined. A similar procedure is followed for background
radiation, and the results of the two studies are compared.



Procedure: Place the 60Co source on the window of the PASCO Geiger tube. Using Data Studio,
take a series of at least 100 measurements of the number of counts using a time interval of one-half
minute. Place the 60Co source back in the lead brick housing. Take a series of at least 100
measurements of the number of room background counts using the same time interval. In the same
manner, take a very low background count rate by using lead bricks to shield the detector as
completely as possible.


Data Analysis: Copy the Data Studio data to Excel or KaleidaGraph. For each of the three data
sets, determine the sample average, A , for the 60Co source and the exact standard deviation σ. In
each case compare the actual standard deviation with the approximation σ ≈ A . Do you find
agreement? (Recall again that you have a limited sample of measurements.)
                          €
                                                                     €

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    Physics 340                                                                  Spring (3) 2010

    Construct a frequency distribution curve for each data set. You will need to bin your data to create
    these plots. You have to do this manually in Excel. In KG you can do this using the "bin data"
    function.

    Using the graph for each data set, determine the fraction of data points that are within the range
    A ± σ . Make similar determinations for the fraction of measurements within the ranges A ± 2σ and
    A ± 3σ . How do these fractions compare with the expected values for a Gaussian distribution?

    Compare each experimental frequency graph with a plot of the theoretical Gaussian (for all data sets)
€                                                                               €
    and Poisson (for background data only) distributions by plotting these distributions on the graph
€
    along with your data (as done in Fig. 1). The Gaussian distribution should be symmetric while the
    Poisson distribution should be skewed slightly toward the lower counts. Do any of your frequency
    curves indicate this? What do you conclude?

    Do the counts in the second portion of the lab need to be corrected for coincidence (i.e., dead-time)
    losses? Why or why not?




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