Re: Nuclear rate of decay quandary

["Steven T. Ratliff" <str@NWC.EDU>]

Regarding Donald Simanek's questions:

> It would be an interesting problem for the rest of you to tackle.
> Based on this data, estimate the uncertainty in the determination of
> half-life. Also estimate the uncertainty of that answer of 128 atoms
> left after 6 minutes. Assume the uncertainties in the
> instrumentation, including the times, are negligible.

There were 1024 atoms originally.  The original question said that
these decayed to 1 atom after 20 minutes.

The key fact that I use is that we are really counting
disintegrations (not the number remaining) and that the uncertainty
in the number of disintegrations is the square root of the number.
The disintegrations are occuring randomly, but they have a
constant average rate (over long times).

For the first question, having one atom left means that we counted
1023 decays.  You can write a formula for the half life in terms of
the number of atoms that decayed.  Let T be the half life
 Let N1 be the number of atoms decayed = 1023
delta N1 is the uncertainty in the number of atoms decayed =
sqrt(1023)

delta T is the absolute uncertainty in the half life.

Let N0 = the number of atoms at t= 0, or 1024 atoms

Let t = the time for the experiment = 20 minutes

T = - t ln(2) /(ln(1-N1/N0))

Using standard error propagation theory:

delta T = absolute value of (dT/dN1) times delta N1

Performing the differentiation and substituting the numbers, I get

delta T = 9.229 minutes , or approximately 9 minutes.

The uncertainty in the half life, 9 minutes, is greater than the
half-life (2 minutes), indicating that, as Donald said, the
experiment is not really valid as a means of finding the half life.

For the second question, there are two answers.  The answer to the
first question shows that the calculated half-life cannot really be
used.

If you ignore the error in the half life, and just look at the
errors in counting, we should report that after 6 minutes,
we have 128 plus or minus 30 atoms left (this is based on statistical
errors only, not the error in the half-life).  The error in half-life
appears to be so large as to make calculations meaningless.

Steven Ratliff




Steven T. Ratliff
Associate Professor of Physics
Northwestern College
3003 Snelling Av. N.
Saint Paul, MN  55113-1598

Internet:  stratliff@nwc.edu (or str@nwc.edu)