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\section*{Introduction to numerical projects}

Here follows a brief recipe and recommendation on how to write a report for each
project.
\begin{itemize}
\item Give a short description of the nature of the problem and the eventual 
numerical methods you have used.
\item Describe the algorithm you have used and/or developed. Here you may find it convenient
to use pseudocoding. In many cases you can describe the algorithm
in the program itself.

\item Include the source code of your program. Comment your program properly.
\item If possible, try to find analytic solutions, or known limits
in order to test your program when developing the code.
\item Include your results either in figure form or in a table. Remember to
       label your results. All tables and figures should have relevant captions
       and labels on the axes.
\item Try to evaluate the reliabilty and numerical stability/precision
of your results. If possible, include a qualitative and/or quantitative
discussion of the numerical stability, eventual loss of precision etc. 

\item Try to give an interpretation of you results in your answers to 
the problems.
\item Critique: if possible include your comments and reflections about the 
exercise, whether you felt you learnt something, ideas for improvements and 
other thoughts you've made when solving the exercise.
We wish to keep this course at the interactive level and your comments can help
us improve it.
\item Try to establish a practice where you log your work at the 
computerlab. You may find such a logbook very handy at later stages
in your work, especially when you don't properly remember 
what a previous test version 
of your program did. Here you could also record 
the time spent on solving the exercise, various algorithms you may have tested
or other topics which you feel worthy of mentioning.
\end{itemize}



\section*{Format for electronic delivery of report and programs}
%
The preferred format for the report is a PDF file. You can also
use DOC or postscript formats. 
As programming language we prefer that you choose between C/C++ and Fortran90/95.
You could also use Java or Python as programming languages. 
Matlab/Maple/Mathematica/IDL are not allowed as programming
languages for the handins, but you can use them to check your results where possible.
The following prescription should be followed when preparing the report:
\begin{itemize}
\item Use Classfronter to hand in your projects, log in  at 
blyant.uio.no and choose 'fellesrom fys3150 og fys4150'.
Thereafter you will see an icon to the left with 'hand in' or 'innlevering'.
Click on that icon and go to the given project. 
There you can load up the files within the deadline.
\item Upload {\bf only} the report file and the source code file(s) you have developed.
The report file should include all of your discussions and a list of the codes you have developed. 
Do not include library files which are available at the course homepage, unless you have
made specific changes to them.
\item Comments  from us on your projects, approval or not, corrections to be made 
etc can be found under
your Classfronter domain and are only visible to you and the teachers of the course.
 
\end{itemize}
Finally, 
we do prefer that you work two and two together. Optimal working groups consist of 
2-3 students. You can then hand in a common report. 

\section*{Project 2, deadline  3 october  12pm (midnight)}


We assume that at $t=0$ we have  $N_X(0)$ nuclei of the type $X$
which can decay radioactively. At a given time  
$t$ we are left with 
$N_X(t)$ nuclei. With a transition rate $\omega_X$, 
which is the probability that the system will make a transition to another 
state during a time step of one second, we get the following differential equation
\begin{equation}
   dN_X(t)=-\omega_X N_X(t)dt,
\end{equation}
whose solution is
\begin{equation}
   N_X(t)=N_X(0)e^{-\omega_X t},
\end{equation}
and where the mean lifetime of the nucleus $X$ is
\begin{equation}
   \tau =\frac{1}{\omega_X}.
\end{equation}
If the nucleus $X$ decays to  $Y$, which can also decay,
we get the following coupled equations
\begin{equation}
   \frac{dN_X(t)}{dt}=-\omega_XN_X(t),
\end{equation}
and
\begin{equation}
   \frac{dN_Y(t)}{dt}=-\omega_YN_Y(t)+\omega_XN_X(t).
\end{equation}

We assume that at  $t=0$ we have $N_Y(0)=0$. 
In the beginning 
we will have an increase of $N_Y$ nuclei, however, they will decay thereafter.
In this project we let the nucleus $^{210}$Bi represent $X$. It decays 
through $\beta$-decay to $^{210}$Po, which is the $Y$ nucleus in our case. 
The latter decays through emision of an $\alpha$-particle to
$^{206}$Pb, which is a stable nucleus.
$^{210}$Bi has a mean lifetime of  7.2 days while $^{210}$Po 
has a mean lifetime of  200 days. 

\begin{enumerate}
\item[a)] Find analytic solutions for the above equations assuming 
continuous variables and setting the number of 
$^{210}$Po nuclei equal zero at $t=0$.
\item[b)] Make a program which solves the above equations.
What is a reasonable choice of timestep $\Delta t$? 
You could use the program on radioactive decay from the web-page of the 
course as
an example and make your own for the decay of two nuclei.
Compare the results from your program with the exact answer as function
of $N_X(0)=10$, $100$
and  $1000$. Make plots of your results.
\item[c)] When $^{210}$Po decays it produces an $\alpha$
particle. At what time does the  production of $\alpha$ 
particles reach its maximum? Compare your results with the analytic ones
for 
$N_X(0)=10$, $100$
and  $1000$. 
\end{enumerate}


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