The Muon Decay Experiment - a
simulation
The historical experiment upon which the present simulation is based was
conducted by Rossi and Hall in 1941. The essence of the experiment was to
measure the number of muons arriving at their detector as a function of position
in the Earth's atmosphere. Muons are elusive subatomic particles that are
created when cosmic rays hit air molecules high in the upper atmosphere. Muons
are highly unstable and, when produced in the lab on Earth, decay with a half life
of 2.2 x 10-6 s (micro-seconds). They provide us with a very precise natural clock!
In the original experiment, the flux of muons was measured at a point near the
top of Mount Washington, New Hampshire at about 2000 m altitude and also at
the base of the mountain. Muons are steadily decaying as they travel down to
earth, and knowing the muon's half life, it is quite easy to predict how many
muons should reach the base of the mountain if we know how many are being
detected at a point 2000 m higher up. Rossi and Hall found the muon flux at the
base of the mountain was much higher than expected - suggesting that these
muons were living longer! When the time dilation relationship was applied, the
result could be explained if the muons were traveling at 0.994 c.
In the "Lab Frame"
To begin, let's gain some experience with the idea of half-life and how this will
relate to our study of muons. Discuss and answer the following:
1. In your own words, explain what is meant by the term half-life
2. Suppose you have prepared a sample of 1000 muons. How many muons
will remain:
a. after 2 half-lives
b. after 14.6 micro-seconds?
3. A sample that originally contained 1000 muons now contains only 38. How
old is the sample?
4. If muons are produced 90 km above the earth, and assuming that they
travel 0.999c, how much time is required for the muons to reach ground
level? Express your answer in:
a. micro-seconds
b. half-lives
5. If muons are being produced at a rate of 1000/s at 90 km above sea level,
estimate the maximum number of muons that should survive the trip
ignoring any corrections for time dilation.
6. Derive the following formula:
where  is the half-life,
No is the original number of muons in the sample and N(t) is the number of
muons after time t. (Hint:
)
7. The following table provides decay information for a sample of muons.
Plot this data in EXCEL and fit an exponential trendline and from this
determine the half-life for the muon.
t (mu s)
0
1
2
3
4
5
6
7
8
9
10
11
N(t)
1010.00
742.56
559.38
422.77
301.33
218.83
160.38
136.86
84.45
75.33
59.36
31.99
The Simulation
1. Our
simulation
allows us
measure the
muon flux by
moving a
probe to
various
heights in
the
atmosphere.
Do this by
clicking on
the probe
and
dragging.
(Press help
on the applet
for more
information.)
2. To make a
reading
press the
start button
(
found
bottom left)
and then
press the
stop button
(
).
3. The muon
flux will be
measured
and
recorded for
you.
4. Start at an
altitude of
150 km and
make
measureme
nts in 10 km
steps (finer if
you wish) as
you move
down to
ground level.
5. When you
have
collected
your data
press the
graph button
(
) then
right-mouse
click in the
graph panel
and choose
the generate
table option
to export
your data
into EXCEL.
image only - click here to load the applet on a separate page
Procedure and Analysis
1. You will need to re-arrange your data to enable you to put it into a table
similar to the one given in question 7 above. To do this you will need to
estimate the time taken for the muons to travel from 90 kms to the heights
at which you measured the muon flux. Since we expect that the muons
are moving at nearly the speed of light we can estimate this time by simply
adopting "c" as the velocity of the muons. Prepare a table that looks like:
Distance Travelled (from 90 km)
0
10
Time (micro seconds)
-----
Muon Flux
-----
2. Plot a curve and trendline for your data and from this determine the halflife of the muons that you measured with the probe.
3. You should get a very different half-life for the moving muons than for the
lab frame muons (2.2 micro-seconds). Compare the two half-lives; which
is "longer" and by what factor?
Questions
1. Explain how your data illustrates the equation
where
2. What value for gamma does your measurement for the half life of the
moving muons imply?
3. Use your value for gamma to determine the speed of the muons.
Literature links
http://www.heliwave.com/gaasenbeek/spap5.html
http://www.marxists.org/reference/archive/einstein/works/1910s/relative/index.htm