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Re: Geiger (a challenge)

LK!

Damn!

You got the jump on me. note following msg. (I was going to ask you, LK, if the Vernier applican. could record in addition to arrival times, event intervals.

Dear Bernard,
I have been incredibly busy lately, so I only now read your email.
Sorry! I will take a look again when I can use my math resources
in my office to see if I can give any guidance. However,
I am leaving on Thursday for a trip to Washington, then
a week from Thursday I go to Brazil for a month and then
to Japan for four months. So many details to address first!!
Best wishes,
Dave

On Wed, 5 Jul 2000, Bernard G. Cleyet & Nancy Ann Seese wrote:

Dave!

I spend too much time on the phys list! here's a ? that I've developed. And being maths
illiterate, I need help. I've noticed that random events come in pairs or small bunches -- now I've
read that
the P. distrib. predicts this. Being an expt'ist I checked it using a G-M counter, a* recorder,
and a ruler. Sure enuff. the freq. of intervals between each pulse does follow (by eye) exp-t**. Now
I
want to cf. my data with the formula. I'm unable to find the mean (int.: t(exp-(rt)r)dt -- zero
to T or inf. My data indicates >65% of the intervals are shorter than the mean interval. Anyway,
I'd like to find (closed form) the mean and the second moment too (sigma sqrd.). And I suppose, to
compare, I need to normalize the fcn. too.

bc

P.s. I tried int: r(exp-(rt)) = -(exp-(rt))(rt+1)/(r*r) and got lost.


* X-Y using its time base
** time between counts

The reason I wanted to do this was to detmn. if data taken with the Vernier system was valid as a source of data for LK's paper. I thought it might not because a sig. dead time would skew the moments
-- The mean would tend to be longer because only one of a pair of counts with a short interval would be recorded and the variance would be narrowed for the same reason. Evans gives the formula for
the corrected variance as a fcn. of the dead time and the counted rate. Melissinos derives the interval probability function from the Poisson probability. In searching for the answers to my
questions above I found: http://www.math.uah.edu/stat/poisson/poisson1.html

Returning to the beginning of the preceding paragraph, I was (still am) concerned because LK had sent me data to calculate the dead time using the two source method. My calc. was ~ one millisecond!
I think that's high enuff to bring into question the validity of his source of data. However, I didn't notice in skimming the paper any flag that this had happened. I've spent the last few days in
addition to collecting data for r(exp-(rt), a search of two source dead time formulae. The best one I found was in the text I used in my rad. course at UCSC in '58! I found that at only rather
high counting rates (> ~ 12K cpm) may one ignore including the bkgnd. So I tried using a similar tube and counting rate (w/o bkgnd) to LK's. I also obtained ~ one millisec.; using the bkgd. formula,
~650 micro sec. (would some one out there write to me how to write 1E-3 & 1 E-6 second?)

Note that LK's mean, 1.8k cpm, assuming a DT of ~ 0.7 milliS. results in an ~ counting loss of 2%. [N/m = 1/(1-mt) m -- detected cpm; N -- "actual" cpm; t -- "dead time"] This assumes a
symmetrical distrib. about the mean -- not so; the point of "our" exercise.


Finally over the past several weeks + I've found for three diff. (type and mfg) G-M tubes a large variation of DT with counting rate. I don't know whether this is an artifact of an incorrect
(incomplete) DT formula or a characteristic of G-M tubes -- Evans does write that this is "thought to be so." -- I hope to make a very low DT tube (some day) using multiple, closely spaced anode
wires. (Patented by R. Eisberg in the 50's). The exptl. way of determination. Perhaps one of you may answer this question, theoretically. [ for the LND 7232 tube: one K CPM -850 microS.; 3K, 650;
5k, 450; 8k, 350; 16k, 310; 24k, same; 100k, 225 -- count rate of both sources, using bkgnd. formula and and sigma only a few %. unfortunately determined by using an approxamation formula for the
dead time -- using a more exact one would be "very hirsute!"]

Comment(s) on the below:

Not unusual -- Evans and Melissinos discuss it and Melissionos gives student data collected from background.

The general interval formula is r ( (rt) ^m-1 ) ( e^-rt ) / ( m-1)! m is integral -- the number of counts between which the time intervals are measured. for m = 1 there is no max. and is the LK
process (interval between each count) . for m > 1 there is a max and the intervals become symmetrical about the mean (limit). Those oldies having used a scale of 64 and a mechanical register tapped
at the output of 4, 8 16, etc. will remember that this progressive symmetrization is very audible!

I now read that Vernier's application does not give interval times (I read somewhere something to the effect that a Geiger-Müller counter is actually an interval counter).

Seems to me the funeral in not - not valid. I wonder if counting near a traffic light would invalidate the poisson process. something to do with CLT or am I "all wet."


Ludwik Kowalski wrote:

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).

arrived waited
seconds seconds
------------------------
0.221
0.288 0.067
0.337 0.049
0.718 0.381
0.920 0.202
1.017 0.097
1.153 0.136

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.)

cnt/0.2 seconds probability
0 ?
1 ?
2 ?
3 ?
4 ?
5 ?
6 ?
7 ?
8 ?

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