Here is an example of an unusual experiment for
your lab. It has to do with statistic of counting.
And there is a challenge for you at the end. The
activity can be set up with the following equipment:
1) Geiger counter ($145, part SRM-BTD, Vernier.)
2) LobPro interface ($220, part LABPRO*, Vernier.)
3) Mac or PC computer
4) A radioactive source such as Co-60 or Cs-137
* An older ULI for Mac or ULI for PC can be
used instead of the LabPro box.
**************************************
The Geiger counter connected to a Mac (via ULI)
was recording times at which particle were arriving.
The table below shows a sample from a long file. The
first column shows times at which arrivals of particles
were recorded, the second shows time intervals between
consecutive recordings (waiting times).
I took 3000 numbers from the second columns and
constructed the histogram. Here are the results:
(0.00 to 0.20 s) dt=0.1 Bin #1, --> 1202
(0.20 to 0.40 s) dt=0.3 Bin #2, --> 726
(0.40 to 0.60 s) dt=0.5 Bin #3, --> 403
(0.60 to 0.80 s) dt=0.7 Bin #4, --> 266
(0.80 to 1.00 s) dt=0.9 Bin #5, --> 154
(1.00 to 1.20 s) dt=1.1 Bin #6, --> 100
(1.20 to 1.40 s) dt=1.3 Bin #7, --> 56
(1.40 to 1.60 s) dt=1.5 Bin #8, --> 34
(1.60 to 1.80 s) dt=1.7 Bin #9, --> 19
(1.80 to 2.00 s) dt=1.9 Bin #10, --> 11
This distribution of inter-arrival times fits
the y=1540*exp(-2.56*x) curve. For example, for
bin #1 the curve gives 1191, for the bin #2 it
gives 714, and for the bin #5 it gives 154. Do not
confuse this exponential curve with the usual
decay curve; the half-live of the source (many
years) is much larger than the time of several
minutes during which these data were collected.
Numbers of occurrences (last column) were converted
into approximate probabilities (dividing each by 3000).
Thus the probability of dt=0.1 was 0.400, the
probability of dt=0.3 was nearly 0.242, etc. The
average waiting time (the inter-arrival time) turned
out to be 0.325 s, the standard deviation 0.155 s.
The exponential distribution may be counter-intuitive.
Knowing that the distribution of counts per unit time
is Gaussian (or Poissonian, to be more general) many
of us would expect a bell-shaped distribution of dt.
For example, at the average counting rate of 5 counts per
second one may expect that distribution to be symmetrical
about the mean time of 0.2 seconds. The experiment
contradicts such expectations.
Note that according to my table 5 particles were recorded
in the first second. In the next second it can be only 3,
or as many as 7 or so. Here is a challenge. Knowing the
above exponential distribution of waiting times predict the
distribution of counting times. In other words, complete
the table below. (Hint: the distribution of cnts/0.2 s
is Poissonian.)
Explain your method. Experimentally determined
probabilities will be posted in two or three days.
***************************************
PS: It is tricky experiment to setup. I would
not be able to do it without John Gastineau helping
me. What he had to do was to process Geiger data
as if they were photogate data. He had to modify
an existing LoggerPro routine. I am trying to
persuade Vernier to include the routine into the
next version of LoggerPro.
But they will do it only if there is a good chance
that many people would use the activity. (LoggerPro
contains a large collection of routines for common
experiment, such as motion detector or smart pulley.
The current routine for the Geiger probe shows the
histogram of counts, for example, ctns/0.2 seconds,
if you want, but not the histogram of waiting times
and not the table of recording times.
If you feel that data on particle recording times
are worth collecting then please endorse this request
on Phys-L. Also share other ideas you may have on
how else can recording times be used in a lab. I think
that probability and statistics teachers may find the
Geiger counter to be a unique source of clean data.
[Sure you can count cars instead of particles. But it
takes much longer and data may be contaminated by
effects of hidden variables, for example a funeral
or an accident somewhere along the road.]
Ludwik Kowalski