\input{labpreamble}
\begin{document}
\subsubsection*{PHSX 101L\hfill Experiment 15\hfill Name:\underline{\hspace*{2in}}}

\underline{Title:} Absorption of Gamma Rays

\underline{Object:} To determine how aluminum absorbs gamma rays.

\underline{Introduction:} The radioactive source used to provide gamma
radiation for this experiment is a 5.0~$\mu$Ci cesium-137 sample.  This
isotope has a half-life of 30.2 years and emits gamma ($\gamma$)
radiation at an energy of 0.6616 MeV. It also emits a
some beta minus ($\beta^{-}$) radiation which we don't want.

It has been found that gamma radiation interacts with matter in a
manner quite different than alpha or beta rays (particles).  This is
due to the fact that gamma rays consist of high energy
electro-magnetic waves, while alpha and beta rays are made up of
high-speed charged particles.  As alpha and beta particles travel
through matter they scatter off the atoms and thus follow a rather
zig-zag path through the material, slowing slowing down as they go.  A
gamma ray, being an EM wave or photon, can travel at only one
speed---the speed of light.  High energy photons can travel through
significant amounts of material without interacting with the matter,
but the greater the distance of travel through a substance, the
greater probability that the photon will be absorbed.  When they do
interact they can lose their energy all at once, rather than in bits 
and pieces..

The absorption process can be described mathematically as
``exponential decay as a function of distance.'' This means
that the number of photons (counts) per minute $N$ that survive a distance $x$ in
a substance is given by the formula
\begin{equation}\label{N}
N = N_{o}e^{-\mu x}
\end{equation}
where $\mu$ is the linear absorption coefficient of the material, and $N_{o}$ is the initial
count rate. Solving equation~\ref{N} for
$\mu$ gives
\begin{equation}\label{mu}
\mu = {-\ln ( N/N_{o} )\over x}
\end{equation}
where $\ln$ is the natural (base e) logarithm (you should have a $\ln$ 
button on your calculator).

A useful number related to this absorption process is the 
\underline{half-value layer} (HVL), also known as the half distance, or $x_{1/2}$ of a substance.  
This is defined to be the distance through 
the material for which one half of the original photons will be 
absorbed.  It can be calculated from equation~\ref{mu}.
\begin{equation}\label{x}
x_{1/2} = \frac{-\ln\frac{1}{2}}{\mu} =  \frac{\ln 2}{\mu} \approx \frac{0.6931}{\mu}.
\end{equation}
\noindent
So, once we have determined the absorption constant $\mu$, we can
easily calculate the half-value layer $x_{1/2}$.

Because the decay process is totally random and predictions can be 
made only statistically, it is important to get as many counts as you 
have time for.  The effect of the air is negligible in this experiment.


\underline{Apparatus:}  Draw a diagram of the apparatus.
\vspace{1.4in}

\underline{Safety:}  The radioactive sources are of extremely low 
activity and are not hazardous when used as directed.  However, it is 
a good idea to treat all radioactive materials safely.  For example, 
do not put the source in your pocket or bring it too near your body; 
handle it only when necessary, and return it when the experiment is 
over.  For the safety of the equipment, please do not touch the end 
window of the Geiger tube or you will break it and destroy the tube.  
It cannot be repaired and it is expensive to replace.

\underline{Procedure:} It would be convenient to make all time 
measurements in minutes for this lab.
\vspace{-.25in}
\begin{enumerate}

\item With the Geiger tube mounted in its stand and with all 
radioactive sources placed well away, measure the background 
radiation.  To do this simultaneously press the reset button on the 
counter and start your stop watch.  Allow the counts to go to at least 
200 and record the time (an integer number of minutes is convenient). 

\item Place the $^{137}$Cs source, printed side up, in the depression
at the bottom of the plastic stand.  (And place one aluminum plate in
the stand.  This is done to block out the alpha and beta radiation
that may also be emitted by the source.  These rays are easily blocked
by a single plate of aluminum, whereas most of the gamma radiation
passes through.  Do not count this beta-shield in your plate thickness
measurements.)

Simultaneously, press the reset button on the counter and
the start button on the stopwatch.  Stop the stopwatch when the
counter reads at least 3000 counts.  Then correct the count rate for
the background radiation by subtracting the background count rate. 
This corrected count rate is $N_{o}$.  It will be the first entry in 
the corrected count rate column of your table (corresponding to 
$x=0.00$~cm).

Then insert additional aluminum plates one at a time noting the cumulative
plate thickness and the corresponding counts each time; let the counts
go to at least 1000 (more is better) for each thickness.  Then correct
the count rate for the background radiation by subtracting the
background count rate.

% \item Measure the thickness of four additional plates of aluminum and 
% add them together.
% 
% \vspace{.1in}
% \underline{\hspace{1.0in}}\hfill 
% +\hfill\underline{\hspace{1.0in}}\hfill +\hfill
% \underline{\hspace{1.0in}}\hfill 
% +\hfill\underline{\hspace{1.0in}}\hfill =\hfill
% \underline{\hspace{1.0in}}
% 
% \vspace{-.1in}
% \hfill total $x$\ \ \ \ \ 
% 
\item Make a graph of $-\ln(N/N_{o})$ vs.~$x$.  From equation
\ref{mu}, we see that the slope of the best-fit straight line graph
should be $\mu_{\rm Al}$.

\item Record your slope as the experimental value of $\mu_{\rm Al}$
and use equation \ref{x} to compute your experimental value of the
half-value layer $x_{1/2}$.

\item Interpolate the accepted value of $x_{1/2}$ from the chart on the 
web at 
\newline\verb=<http://www.safety.queensu.ca/safety/radiation/course/hvl_chart.htm>= 
and compare your value to it.

\end{enumerate}

\newpage
\underline{Results:} 
\vspace{-.25in}
\begin{enumerate}

\item 
Counts = \underline{\hspace{1.in}}\hfill Time = \underline{\hspace{1.in}} 
\hfill Background count rate = \underline{\hspace{1.in}}
\vspace{.1in}

\item \ %(Remember that $N_{o}$ is the corrected count rate when $x = 0.00$.)

\vspace{-.2in}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}                                                                   \hline
total plate &               &                      &            & corrected  &               \\  
 thickness  &\ \ counts \ \ &\ \ \ \  time \ \ \ \ & count rate & count rate &               \\  
    $x$     &               &          $t$         &            &     $N$    &$-\ln(N/N_{o})$\\
    (cm)    &               &         (min)        &(counts/min)&(counts/min)&               \\  \hline%\hline
    0.00    &               &                      & &$N_{o}=$\ \ \ \ \ \ \ \ \ \ \ \ \ &0.00\\  \hline
            &               &                      &            &            &               \\  \hline
            &               &                      &            &            &               \\  \hline
            &               &                      &            &            &               \\  \hline
            &               &                      &            &            &               \\  \hline
            &               &                      &            &            &               \\  \hline
\end{tabular} 
\end{center}
\vspace{.1in}

\item \ %Graph of $-\ln(N/N_{o})$ vs.~$x$.

\vspace{-.2in}
\setlength{\unitlength}{.1mm}
\begin{picture}(1700,1200)(-20,0)
\multiput(0,0)(100,0){18}{\line(0,1){1200}}
\multiput(0,0)(0,100){13}{\line(1,0){1700}}
\end{picture}

\vspace{.3in}

\item
$\mu_{\rm Al} = $ \underline{\hspace{1.2in}}\hfill Experimental $x_{1/2} $  =  \underline{\hspace{1.2in}}

\vspace{.2in}
\item
Accepted value of $x_{1/2} $ = \underline{\hspace{1.2in}} \hfill \% diff =
  \underline{\hspace{1.2in}}


\end{enumerate}

% http://www.safety.queensu.ca/safety/radiation/course/hvl_chart.htm
% 
% Tutorial at <http://www.bh.rmit.edu.au/mrs/DigitalRadiography/DRPapers/DEXA_Attenuation.html>
% 
% Interaction of gamma with matter 
% <http://www.triumf.ca/safety/tsn/tsn_6_3/subsection3_5_2.html>
% <http://www.dne.bnl.gov/CoN/nuc/C/Co60.shtml>
% <http://www.doegjpo.com/programs/hanf/BYReport/Report/sect3.htm#3.2>

\newpage
\underline{Questions:} 
\vspace{-.25in}
\begin{enumerate}
	\item Explain why if your data points do not lie well on a straight line.
\vspace{.7in}
	\item In introductory physics we have gone to great lengths to make 
	our graphs linear (you linearized some graphs in the Simple Harmonic 
	Motion lab by squaring the periods).  This is because the math of 
	straight lines is straight forward.  You linearized your graph today 
	by taking logarithms.  Qualitatively sketch a graph of 
	your data \emph{without} linearizing it, {\it i.e.,} a graph of $N/N_{o}$ 
	vs.~$x$.  No numbers are necessary.
\vspace{1.2in}
	\item Average your value of $x_{1/2} $ with those of the other 
	groups in the class.  How much closer is the average to the accepted 
	value?  Discuss systematic vs. random errors.
\vspace{.7in}
    \item By what factor would you expect the count rate to fall off 
    if you put four times the half-value layer of some material 
    between the source and the Geiger tube?
\vspace{.5in}
	\item If you start with $8\times 10^{20}$ atoms of $^{137}$Cs, about how 
	many would your grandchildren have left (undecayed) 90.60 years later?  (Hint: how 
	many half-lives is 90.60 years?)
\vspace{.5in}
	\item By what factor would you expect the count rate to fall off 
	if you move the Geiger tube twice as far away from the source 
	(without any shielding plates)?
\vspace{.5in}
\end{enumerate}

\underline{Conclusions:} 
\vspace{-.25in}


\end{document}
