• Study Resource
• Explore

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Central limit theorem wikipedia, lookup

Transcript
```Radiation Experiment 1
Prelab questions for this exercise can be found on Page R-11
Safety
Make sure that you have read the safety notes in the introductory section of this manual before
beginning any practical work.
Do not, under any circumstances, attempt to repair any of the equipment. Rather, turn off the
apparatus at the power point and consult your demonstrator.
While it is a general rule that you may not eat or drink in the laboratories, it is especially
emitters of -radiation can be very dangerous.
If you need a drink go to the fountain at the eastern end of the laboratories.
general precaution applies regardless of the strength of the sources. To do this you should try to:


minimise the time that you are exposed to the radioactive source

shield the sources wherever possible (use the lead blocks provided to do this)

wash hands after the laboratory session.
Paradoxically, the lead that is used to shield the radioactive sources is also toxic. For this reason
it is doubly important that you wash your hands after the laboratory session (especially before
lunch).
The radiation detectors used in this laboratory are powered by a high voltage power supply. As a
result of this you should not re-wire any of the circuit modules without consulting a demonstrator.
If any of the components or leads are damaged these should be brought to the attention of the
demonstrator.
Outline
In this experiment you will learn about the random nature of radioactive decay and measure a
fundamental property of all radioactive nuclei, the half-life. You will establish whether your data
follows a “normal distribution” by comparing it with theoretical expectations of this distribution.
First you will use a computer simulation of a normal distribution and examine its various
properties to familiarise yourself with its behaviour. The concept of percentage statistics is
introduced. The techniques you explore here will be used in the next experiment to estimate
uncertainties in the data you collect from radioactive samples.
Secondly you will measure a half-life, which is one of the fundamental properties of a radioactive
substance. You will plot your data, estimate uncertainties and finally fit a straight line in order to
quote a value for the half-life. By plotting lines of maximum and minimum gradient you should be
able to quote uncertainties for your result.
Physics 121/2 and 141/2 Laboratory Manual
R-5
References
121/2: Sections 45.3, 45.4, 42.3.
141/2: Sections 43.3 and 43.5
Section A
Normal Distribution
Introduction
Radiation is emitted in a random way from radioactive materials. You cannot know when a
particular nuclide will decay only its probability of decay. In spite of this, a collection of
radioactive nuclei does emit radiation in a regular way that can be reasonably well described by a
normal distribution.
It is quite likely that you are already familiar with normal distributions, even if your knowledge
developed in an informal way. In its simplest form two quantities, the mean and the standard
deviation can describe a normal distribution. For example I might want to count the number of
cars which enter the university between 9:00 AM and 10:00 PM on a Monday. Presumably this
number would change a bit from week to week, but is probably roughly constant. If I were to
collect this data over a series of weeks I would be able to tell you two things. The mean number of
cars would simply be the average count. The standard deviation would be a number, which I could
calculate, that would tell you about the spread of results. It would give an idea of how close you
could expect an individual count to be to the mean of all counts.
This sort of treatment is very useful when measuring something which is not perfectly regular,
but remains “roughly” regular. We could represent the occurrence of such data as a histogram (as
shown in figure 1 below). The theoretical distribution might look as shown in figure 2. Here the
area under the curve between x = a and x = b represents the number of observations that have
values between x = a and x = b.
Frequency of X
Frequency of X
X
a b
Figure 1
X
Figure 2
If we take N measurements of some quantity (let’s say x), then the mean, , is defined by
N
 =
x
i
i=1
N
,
and the standard deviation, , is defined by,
N
 =
R-6
(x
i
-)
i=1
N -1
2
.
Physics 121/2 and 141/2 Laboratory Manual
Properties of the normal distribution — exploration using StatPlay
A1:
Double click on the StatPlay icon in the labtools group on your computer. You will explore how the
quantities mean and standard deviation describe the distribution. Choose the continuous
distribution explore window. You should now see a normal distribution with a mean of 0, and a
standard deviation of 1.
Exercise 1 – Exploring  and 

Describe the effects on the distribution of moving the mean. This can be achieved by
dragging the box labelled . Small labelled sketches will be the most effective way of

Now, in a similar way, describe the effects of changing the standard deviation (achieved by
dragging the box labelled ).
From this distribution of values about the mean, it is possible to tell the likelihood (probability) of
a single measurement occurring within a particular range of values.
Exercise 2 – Exploring the distribution down to the tail

Change the values for mean and standard deviation to 10 and 1 respectively. Do this by
typing in the values on the right of the screen. Click on the Z-scores button (at the top - it is
the button which shows a graph with vertical lines on it). This will put lines on the graph at
intervals of one standard deviation.

Now look at the probability of taking a measurement that is lower than one standard
deviation below the mean. Do this by clicking on the Left Tail button, which shows a graph
with the left edge coloured differently. By moving this p line about you can record the
probability of a value being below the line. Use this technique to answer the following
questions:
Question (a)
What is the probability of obtaining a value below ?
Question (b) What is the probability of obtaining a value below -?
Question (c)
What is the probability of obtaining a value below -2?
Question (d) What is the probability of obtaining a value below -3?

Now use these measurements to calculate the percentage probability of a single
measurement being within one standard deviation on either side of the mean. Repeat for
the probability of being within 2 and 3 standard deviations of the mean.
These answers should agree with the much used values which your demonstrator will be able to
give you.
Exercise 3 – The median
Another quantity which statisticians will often talk about is the median, which is defined as the
point on the probability distribution below which the probability is one half.

Check your normal distribution to find the point that corresponds to the median. For a
normal distribution the median and the mean should be the same. Compare the mean and
median for the simulated normal distribution and comment.
Physics 121/2 and 141/2 Laboratory Manual
R-7
Properties of the Normal Distribution — the experiment
A2:
Imagine that you have set up a counter to detect the radiation from a long-lived source, that is
one whose activity, the number of particles emitted per second, may be regarded as virtually
constant. You record the number of particles detected in a certain time interval, the count x, and
then repeat a series of measurements for the same time interval. You will find that the counts are
not all equal but are distributed over a small range.
This distribution follows a form known as a Poisson distribution, which can be described by a
single quantity  (the mean), and has a standard deviation, which is equal to the square root of
the mean.
 
That is
Provided  is large enough, the Poisson distribution becomes a normal distribution with the
linking between mean and standard deviation shown by the equation above. This equation is not
true for all normal distributions, only those that are also well described by Poisson statistics.
Your basic aim in this experiment is to firstly observe the distribution obtained from a radioactive
decay and, secondly, to analyse these data to determine whether they do indeed follow a normal,
and Poisson, distribution.
Experiment (i)
Hints on Uncertainties

Whenever we take an experimental measurement we must also consider any errors or
uncertainties associated with it. In this experiment you will need to take into account:

Statistical uncertainties – What is the uncertainty associated with the count rates you
measure? Refer to page R-3 of your manual for help with this.

Background – How do I incorporate background contributions into my error analysis? Refer
to page R-3 and R-4 of your manual for an example of this.

Graphical Errors – How do I apply errors on a graph to a value I read from it? Refer to page
22 of your manual for a discussion of this.
Settings for the Electronics
Before you can begin acquiring data you need to set up the electronics for the experiment. Set up
the apparatus with the settings shown below.
H.V. power supply
Single channel analyser
R-8
900 V

Selector switch to "normal"

Lower level at 4.0 V

Upper level at 10.0 V
Counter
Discriminator at  2-3 V
Amplifier
-ve input and unipolar output
Physics 121/2 and 141/2 Laboratory Manual
Put a 137Cs source in front of the detector and switch on all the electronics for taking data. Ask
your demonstrator to feed the amplifier output into the oscilloscope, and set the gain so that the
largest pulses produced by your source are about 9 V in amplitude.
Do not alter any settings after your demonstrator has set your gain levels.
Record the settings in your logbook.
Collection of Results
Start up the counter program. Enter a name for the file into which you plan to save data – this is
important so that your data is not simply added to the default data file. Set your source at a
distance from the detector that will give about 1000 counts every few seconds. Take 100 such
measurements and save them to your file.
Analysis of Results
Before analysing the data you have just collected you will need to make the data readable by the
StatPlay program. To do this, read the output file from the counter program into Excel. Delete all
the information apart from the actual counts. Save the file as tab delimited text.
Restart the StatPlay program and choose the data distributions explore window. Go to the File
option at the top left of screen and choose open. Remember your file is in the data directory, so
move to there and open up your data. As soon as the data is read you will see three things; the
data listed exactly as it was read in, next to this the data sorted from maximum to minimum, and
then a histogram of the data. Make sure the numbers are as you would expect - if not, ask your
demonstrator to have a look.
The program has not only sorted your data. It has also divided the full range of your data into
about 10 bins and grouped the measurements into those bins. Each data point in the range of the
bin contributes one to the frequency of that bin. These bin groupings are shown on the histogram.
If you click on the first column of the histogram you will see red dots appear next to the data that
is in this “bin”.
Question (e)
Compare the shape of the histogram to the normal distributions observed
earlier. What are the similarities and differences?
Try changing the bin size by clicking on the buttons above the histogram (pictures). Comment on
the change of shape as you change the bin size.
Return to what you consider is the most useful bin size - discuss this with your demonstrator
before proceeding.
Now analyse the distribution and discuss how well it agrees with the ideal normal distribution.
Make use of some of the previously used features of StatPlay (for showing tail probabilities and the
mean and median). Be sure to address the following questions:
Question (f)
Does the median equal the mean?
Question (g) How do the tail probabilities compare to the ones calculated earlier from the
simulated normal distribution?
Question (h) How closely does
  ?
Physics 121/2 and 141/2 Laboratory Manual
R-9
Summary of Section A:

As a summary of the main points learned so far today, describe how you could estimate the
uncertainty in the following situation. Suppose you are conducting an experiment with a
uncertainty in this measurement

Consider an experiment where you are detecting a mean of 50 counts per minute. How long
a counting time would you use to try to obtain an uncertainty of 3% in your result (ie. an
estimated standard deviation that is 3% of the estimated mean)?
Section B
Half Life of Indium
Introduction
If we have a sample of many nuclei of a particular radioactive species we cannot predict when an
individual nucleus will decay. However, all the nuclei have the same probability of decay in a
given period of time. An equivalent way of stating this is to say that each radioactive species has
its own characteristic half-life, ; that is, a time in which half the nuclei in a sample decay
(roughly – remember the randomness). No matter what size sample we start with, after one halflife, half of the nuclei will have decayed and half will not have decayed. A knowledge of the halflife of a radioactive substance allows us to calculate what number of nuclei will be remain
undecayed after time t, provided we know how many there were at the start.
and
1 half-life,
No/2 remain;
2 half-lives,
half of these, ie. No/22, remain;
3 half-lives,
half of these, ie. No/23, remain;
p half-lives,
No/2p remain.
Hence, to find how many remain after time t, we have only to express t as a certain number halflives, such that:
t = p
and the number remaining after time t, Nt, will be:

N0
-t
 N 0 2 1/2
... (1)
-t.ln2 
N t  N 0 exp
  

 1/2 
... (2)
Nt 
2p
and the number which will have decayed,
N t , will be:
Nt (t)  N 0  N t
R-10
-t 

 1/ 2 

 N 0 1  2 


... (3)
Physics 121/2 and 141/2 Laboratory Manual
The decay rate, ie. number of decays per unit time is proportional to the number of nuclei present.
So, if you have two samples of the same radioactive species, with the first having with twice as
many radioactive nuclei as the second sample, then the rate of decay of the first sample will also
be twice the value of the rate of decay of the second sample.
It then follows from equations (1) and (2) that the decay rate, at any time t, is related to the rate
at time zero by the equations:
t
dN 
dN  
1/ 2

 dt 
 = 
dt 
 2
 t
 0
... (4)
dN  -(t.l n2)/
1/ 2
= 
 dt 
 e
 0
... (5)
Equation (4) may be used to work back from measured values of the decay rate, dN/dt, at various
times to find 1/2, and that is precisely what is done in this experiment.
Prelab Questions:
After reading through the Introduction to Section B above as well as the introduction to the
radiation experiments, complete the following questions.
Determining the half-life of 128I
The table below shows some measurements of the decay rate of a sample of
often used medically as a tracer. Background has already been subtracted.
128I,

What would be the statistical uncertainty associated with each of the R values (complete
the table)?

How does background affect the uncertainty associated with R? (Refer to page R-3 and R-4
of the Radiation introduction for help with this).
Suppose the background was 1.6 ± 0.1 counts/second. Calculate the new uncertainties in
each R value.

Sketch a graph of the data (paste this into your manual) and use it to find a value of the
half-life for 128I.
 =
Time (min)
R counts/sec
0
392.2
32
161.4
64
65.5
96
26.8
128
10.9
Physics 121/2 and 141/2 Laboratory Manual
R
R (including
background)
R-11
Experiment (ii)

Reduce the lower level on the single channel analyser to 0.5 V to improve the efficiency of
detection.

Make a measurement of background by counting pulses for intervals of 1 second over a
period of 1 minute.

After the 30-second neutron bombardment, quickly fasten your indium disc to the face of
your scintillation counter with sticky tape.

Beginning immediately, measure the counts from your source for intervals of 1 second,
again for a period of 1 minute.
Analysis of Results

Plot a graph of counts versus time.
Question (i)

What criterion was used to decide when it was no longer worthwhile to collect
data?
From your graph, find the half-life of 116Inm, and express it as:
=
Question (j)
±
seconds.
In plotting your data did you take the time corresponding to each point to be
the start, middle or some other point of the counting period? Is this important
Summing up Section B
R-12
Physics 121/2 and 141/2 Laboratory Manual
A postscript
Natural indium is 4%
113
115
115
49 In and 96% 49 In . When it is irradiated with neutrons, 49 In may absorb
a neutron, producing 116
49 In in one of its excited states. These excited states decay by -ray
emission, leading to either the first excited state or the ground state of 116
49 In , which is a beta
emitter. The first excited state is so different in structure from the ground state that decay from it
to the ground state virtually never occurs. Instead, it decays by beta emission to
life of order 1 hour. The ground state also decays by beta emission to
much shorter—a matter of seconds. It is the short half-life of
116
49
116
50Sn ,
116
50Sn
with a half-
but its half-life is very
In in its ground state that is
measured in this experiment.
To make
we need a neutron source. The californium isotope
116 m
49 In
252
98 Cf
undergoes spontaneous
fission, the products of which emit neutrons. This means that it breaks into two parts of
comparable mass, either or both of which may release free neutrons. It has a half-life of
2.65 years and is the neutron source we use. It is kept in a tube immersed in a tank of water.
The neutrons from the decay of
are slowed down in the water as a result of collisions with
252
98 Cf
the nuclei of the hydrogen atoms of the water (protons). This is done because slow neutrons are
115
49 In
than are fast neutrons. The Indium radioactive sources you will be
using are placed in the tank for about 30 seconds by your demonstrator. This is enough time to
produce the short lived form of
115
In  n 
116
In* 
116In.
The process may be represented by the equation:
Ings   
116
Physics 121/2 and 141/2 Laboratory Manual
Sn   -  
116
R-13
R-14
Physics 121/2 and 141/2 Laboratory Manual
```
Related documents