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Re: Radioactive decay

On this or another list someone suggested drilling holes in a bunch of
dice, throwing them on an OHP, removing the ones that showed the holes,
repeat, etc.

I instead for my demos., purchased 500 dice (at a time when each pkg. of
five included a quarter off cupon, so with the already low <Target>
price they were quite "affordable". Five groups of ~ five students
would throw them and remove the ones (easy to spot the red bicycle TM),
etc. One student tabulated their results on a spread sheet. The
plots (successively averaged) illustrated the smoothing due to increase
in "n". The next day we milked the cow.

Ludwik Kowalski wrote:

In a tangential remark JohnD wrote (yesterday):

If the decay rate of 16.66% per round is too fast, you
can lower it by various means. You can lower by a
factor of 4 by saying that they only decay if they land
black-spot-up _and_ pointing toward the north side
of the room.

This prompted me to go back and to discover a detail
which becomes important when results of simulations
(for large numbers of atoms) are compared with the
experimentally measured, or theoretically predicted
half-life. Here is a challenge for students asked to
create a simulation program, as that shown at the
end of this message (same as yesterday).

I used this program for the initial N=100000 and
generated the following outcomes:

i remaining atoms
---------------------------------
0 100000
1 83400
2 69543
3 57930
4 48347
5 40333
10 16150
15 6373

Then I used another program to obtain the best exponential fit.
The resulting half-live turned out to be 3.73 units. Ask your
students why this result is not the same as the theoretically
expected T=ln(2)/lambda=4.11 units. The answer is that
T=4.11 corresponds to a situations in which atoms decay
continuously and not in 16.6% steps. To get a good agreement
with the theoretical T one must simulate with dN=lambda*N
negligibly small in comparison with N. (dN is the number of
removed pencils per step while N is the total number of pencils)
That is why making lambda smaller is desirable. Note that even
a large lambda, such as 1/2 (using pennies), will also produce
an exponential curve but T will be smaller than what really
happens when lambda is 1/2.

Program Simulation ! In True Basic computer language
randomize ! to avoid exact reproducibilities
let lambda=1/6 ! probability of "decay" per throw
let t=0 ! initialize counter of throws (timer)
let N=10000 ! initialize number of atoms (pencils)
print"At t=0 the number of radiactive atoms was: ";N
for i=1 to 20 ! perform 20 throws
let decays=0 ! initialize counter of decays
for k=1 to N ! for every pencil in a throw
let try=rnd ! a random number (between 0 and 1)
if try<lambda then let decays=decays+1
next k
let N=N-decays ! remaining number of pencils (atoms)
print "after t=";i;" only ";N; "atoms remained."
next i ! ready for the next throw
end
! ********************************************

Ludwik Kowalski