
\subsection{MPC-1b: Absorption of Radiation}\label{mpc01b_absorp}

\noindent {INTRODUCTION}: \begin{enumerate}
\item[] \latex{\vspace*{-.12in}} 
Many atomic nuclei are radioactive.  This means that they spontaneously 
decay, a nucleus of a given atomic number Z (the number of protons in the
nucleus, and the number of electrons buzzing around it) and mass number A 
(the number of protons plus the number of neutrons) transmutes into a nucleus with a different atomic number Z' or a different mass number A' or both.  In this process they emit $\alpha$ particles (helium nuclei), $\beta$ particles (fast moving electrons), or $\gamma$ rays (energetic, penetrating photons).  The instant in time at which an individual nucleus will decay cannot be predicted, this is a random event, but if one observes a large collection of $N$ identical radioactive nuclei, the number $\Delta N$ that decays in a short time interval $\Delta t$ is given by $$\Delta N \approx  -N { \Delta t \over \tau}, \eqno (1)$$
where $\tau$ is the ``mean life" of the radioactive nucleus; it is the time necessary  for  63\% of the original number of nuclei to decay.


If one starts with $N_0$ nuclei at time $t=0$ then the number of nuclei left 
after a time $t$ is:
$$N=N_0 e^{-t/ \tau}$$

The rate $R$ at which nuclei decay is therefore: 
$R = -\Delta N /\Delta t \approx { N  \over \tau} = (N_0/ \tau) e^{-t/ \tau}$

If the mean life $\tau$ is very long compared with the duration of an 
experiment the rate $R$ at which the nuclei decay is practically constant:
$$\Delta N / \Delta t \approx  N_0 / \tau \eqno (2)$$

Often the life of a radioactive nucleus is expressed in terms of a ``half life" $\tau_{1/2}$  which is the time necessary for  50\% of the nuclei to decay; $\tau_{1/2} =0.69 \tau$. The above expression is an  approximation valid when the time interval $\Delta t$ is much shorter than  $\tau$, as is true in this experiment.

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{\em NOTE TO THE INSTRUCTOR:} You should remove the plastic cap from the GM counter for the students studying the absorption of $\beta$ radiation. Please be sure to replace the cap when the lab is finished; or else it will get lost! 
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\noindent{EXPERIMENT} \begin{enumerate}
\item[] \latex{\vspace*{-.12in}}  You will detect and study the absorption in different materials of the decay products of two nuclei, Co$^{60}$ whose 
half life is 5.3 years and which emits $\gamma$ rays of energy about 1.3 MeV, 
and T$\ell^{204}$ whose half life is 3.8 years and which emits $\beta$ 
particles of maximum energy about $0.75 \ MeV$.  The $\beta$ particles have a 
continuous energy spectrum from zero to the maximum. You will find that 
$\gamma$ rays and $\beta$ particle  are partially absorbed as they
traverse a thickness of material, and that the absorption depends 
mostly in the number of electrons per square centimeter in the absorber.

In order to do all this fairly rapidly different groups of students will
do different parts and compare their results at the end.

IMPORTANT:

When a beam of $N_0$ particles crosses a layer of absorber of thickness 
$t \ mg/cm^2$ the number of particles that emerge is given by:
$$ N = N_0 e^{-t/\lambda} $$ 
This formula is similar to the one at the beginning of this writeup: \\
the number of particles absorbed in a small thickness $\Delta t$ is 
$$\Delta N \approx  -N { \Delta t \over \lambda}$$
Here $\lambda$ is the ``absorption thickness'' of the material, and is
it is the thickness necessary  for  63\% of the original number of nuclei to absorbed.

The data has an exponential form; if one takes the logarithm of the
number of particles one obtains a linear expression, which is easier to 
analyze: 
$$\ln N = \ln N_0 - {t \over {\lambda}}$$

\begin{figure}[!h]
\centering \latex{\vspace*{-.25in}}
\includegraphics[height=6.6cm]{figs/l104/fnc1b-2.eps}
\latex{\vspace*{-.15in}}
\latex{\caption{Number of particles vs thickness.}}  
\html{\bf Figure 1: Number of particles vs thickness.}  
\medskip 
\includegraphics[height=6.6cm]{figs/l104/fnc1b-3.eps}
\latex{\vspace*{-.15in}}
\latex{\caption{$\ln N$ vs.~thickness on log-linear paper.  The slope of the line is equal to $-1/\lambda$}}
\html{\bf Figure 2: $\ln N$ vs.~thickness on log-linear paper.  The slope of the line is equal to $-1/\lambda$}
\end{figure}
\end{enumerate}

\noindent {EQUIPMENT} 
\begin{enumerate}   
\item \latex{\vspace*{-.08in}}Geiger-Muller counter$^\dagger$ mounted vertically and stand. This device can detect either $\beta$- or $\gamma$-radiation from the radioactive sources. 
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{$^\dagger$} The Geiger Muller tube was first developed by Hans Geiger(1882-1945) who collaborated  with Sir Rutherford at Manchester in the early work on radioactivity which later led to the discovery of the atomic nucleus by Rutherford.  The counter was then perfected by Muller in 1928 and is now known as the Geiger-Muller (GM) counter.  The counter consists of a fine wire along the axis of a gas filled metal tube.  The wire is made about  $ 500\> V$ positive with respect to the tube.  When ionizing radiation, such as $\gamma$ rays or $\beta$ particles, enters the counter it breaks up (ionizes) a few gas atoms releasing electrons which are rapidly accelerated toward the positive wire.  These electrons collide with other gas molecules, releasing new electrons.  Finally, an avalanche of many millions of electrons reaches the wire and produces a voltage pulse large enough to be detected and counted.  The whole process can be initiated by a single $\alpha$, $\beta$ or $\gamma$ ray entering the counter. 
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\end{tabular}
\item[] \vspace*{-.08in}The Geiger counter is mounted vertically in a stand.
\item \latex{\vspace*{-.08in}} PASCO Interface, receives the impulses from the GM counter
and transfers data to PC
\item \latex{\vspace*{-.08in}} Two radioactive ($\beta$ and $\gamma$ sources).
\item \latex{\vspace*{-.08in}} Lead, Aluminum and Poly absorbers.
\item \latex{\vspace*{-.08in}} A platform with multiple slots.
This permits the radioactive source to be placed at various 
distances from the counter. \\
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\includegraphics[height=7.7cm]{figs/l104/fnc1b-1.eps}\\
\vspace*{-.0in} Figure~3: The Geiger-Muller counter and stand 
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\end{enumerate}

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{\em PRECAUTIONS:}  The Geiger counter has a very thin window to permit the entry of the radiation under study; the window is protected by a plastic cap. Groups working on $\beta$ radiation should ask their TA to remove it carefully.
Do not poke anything up toward the counter.  The window is very fragile, if it is broken the counter will not function, and can not be repaired.
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\noindent{PROCEDURE I: (45 min)}
\begin{enumerate}   
\item[] \vspace*{-.08in }In this procedure you will do the work in separate groups and then discuss your results with the different groups. You will observe the absorption of $\beta$ and $\gamma$ radiation in Lead, Aluminum and Polyethylene:
$\gamma$ and $\beta$ radiations behave very differently in traversing absorbers; because of this your instructions will depend on the source you are studying. 
\begin{itemize}
\item \vspace*{-.05in} $\gamma$ rays are energetic photons, when a $\gamma$ ray interacts with an atom, the $\gamma$ undergoes `Compton Scattering' and is effectively eliminated from the beam. 
\item \vspace*{-.05in} $\beta$ rays are energetic electrons, when an electron traverses matter, it knocks low energy electrons out of the atom, in this way loses energy and eventually stops. \end{itemize}

Because of this the work will be divided among different
groups: \\
\hspace*{1in} Group \# 1 $\Rightarrow $ Poly $\beta$ :
Group \# 2 $\Rightarrow $ Pb $\gamma$ :
Group \# 3 $\Rightarrow $ Al $\gamma$ \\ \\
{\bf NOTE:} {\em All Groups must make a {Background measurement}}
\item[] \vspace*{-.08in }Click on the Launch NC-1B icon below (web version) to initiate the PASCO software window.
The screen should show two windows:  a table window and a graph window.
\input{toolbar/nc1_tb2}
\end{enumerate}
\begin{enumerate}
\item \vspace*{-.03in } Remove the sources from the vicinity of the GM counter, set them at least 1 m away.  
\item \vspace*{-.03in } A total of 300 seconds have been allocated for collecting the data.  If you wish to verify or change this: 1) CLICK on the Setup icon 2) In the Experimental Setup window CLICK on the Options Icon 3) In the Sampling Options window CLICK on the Automatic Stop tab. For group \#1 you may reduce it to 101 second.  For groups \#2 and \#3 the value should remain at 301 seconds.
\item \vspace*{-.03in } By default the time period between data collection points is set to 10 seconds.  To verify this: 1)  CLICK on the Setup icon 2) In the Experimental Setup window DCLICK on the Geiger Counter icon 3) examine the ``Seconds between Samples" setting.  A $10~s$ interval is not really unnecessary but in this way you can see the data begin record without waiting a full five minutes.
\item \vspace*{-.03in } The statistics should appear in the Table window as well.  Multiply the `mean' by the number of time periods (10 or 30 for group  \#1, 30 for groups \#2 and \#3) to obtain the total number of counts. This is how you will measure the total number of counts for all data runs. Record your result in Table I. 

These counts are mostly due to a radiation called Cosmic Rays: energetic protons and other nuclei produced somewhere in the galaxy (probably {supernov\ae}  or pulsars) that impinge on the upper reaches of the earth's atmosphere. These particles interact in the atmosphere and produce secondary particles that reach sea level, cross the roof of the laboratory and finally cross your GM tube. They are mostly electrons, positrons and muons. Their number is small $\sim 10^{-2} /cm^2\cdot s$ but not negligible. This background rate must be subtracted from all subsequent counting rate determinations for the $\gamma$ or $\beta$ 
sources.

\begin{center}{Table I}\\
\vskip 5pt
\begin{tabular}{|c|c|c|c|c|c|} \hline
Absorber name &  $t$      &  Total Counts & $\sqrt{N}$ & $N'=$ & $\ln  N'$ \\ \hline
\hspace*{2cm} & $mg/cm^2$ &  $N$          &   ~~       & $=N-N_{bgr}$ & ~~ \\ \hline\hline
Background  & $0 \ mg/cm^2$  &  ~~        &   ~~       &   ~~         & ~~\\ \hline
~~ &   ~~  &   ~~  &  ~~  &  ~~   &  ~~  \\ \hline
~~ &   ~~  &   ~~  &  ~~  &  ~~   &  ~~  \\ \hline
~~ &   ~~  &   ~~  &  ~~  &  ~~   &  ~~  \\ \hline
~~ &   ~~  &   ~~  &  ~~  &  ~~   &  ~~  \\ \hline
~~ &   ~~  &   ~~  &  ~~  &  ~~   &  ~~  \\ \hline \hline
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\end{enumerate}
\noindent{\em $\gamma$ Pb Group} \\ \vspace*{-.08in } 
\begin{enumerate}
\item[5.] \vspace*{-.08in } $\Rightarrow$ Place two Co sources on top of each other in slot \# 5 (counting from the top), CLICK on the Start icon and wait until the computer stops recording and enter the result in Table I  with absorber thickness $t = 0 \ mg/cm^2$.
\item[6.] \vspace*{-.08in } Place various lead absorbers in the slots between the source and the counter. Record the data in table I. Repeat with different absorbers. ( Use one `T' absorber, then two `T' absorbers and then three`T' absorbers together, for maximum absorption.)
Record your data. \\
NOTE: The standard deviation shown by the computer  is {\em not} the error in the rate measurement because it refers to the number of  time intervals during which the data was taken. The error in the rate measurement depends on the total number of counts, it is $\sigma = \sqrt{N}$. 
%\vfill \eject
\end{enumerate}
\noindent{\em $\gamma$ Al Group} %\vspace*{-.03in } 
\begin{enumerate}
\item[7.] \vspace*{-.08in } $\Rightarrow$
Same as step 5, but put the double source in slot \# 6, CLICK on the Start icon and wait until the computer stops recording. Then enter the result in table I  with absorber thickness $t = 0 \ mg/cm^2$.
\item[8.] \vspace*{-.08in } Place one Aluminum  `P' absorber  in the slots between the source and the counter. Then put two, three and four `P' absorbers in the slots. Record the data in table I. 

ADDITIONAL PRECAUTIONS FOR THE BETA PARTICLES EXPERIMENT:\\
The absorbers for this part are extremely thin.  {\em Please treat them carefully} !!! 
\end{enumerate}
\noindent{\em $\beta$ Poly Group}
\begin{enumerate}
\item[9.] \vspace*{-.08in }  $\Rightarrow$ Check that the plastic cap has been removed from the GM counter. Place the source in slot \#4 with the side without writing facing toward the GM counter. 
%Set the sampling to 101 seconds,  
CLICK on the Start icon and wait until the computer stops recording and enter the result in table I  with absorber thickness $t = 0 \ mg/cm^2$.
%\vfill \eject
\item[10.] \vspace*{-.08in }
Place various Poly absorbers in the slots between the source and the counter. Use a `C', a `D', two `D's, an `E' and a `C' together, an `F', and an `F' and a `D' together. Record the data in table I. Repeat with different absorbers.
Record your data in Table I.

\end{enumerate}

%Using this kind of graph, first one traces a straight line through the data 
%points, then one finds the thickness corresponding to an intensity  
%$N_0 \cdot 1/e = 0.37 N_0$ or $N_0 \cdot 1/2e = 0.14 N_0$  
%or $N_0 \cdot 1/3e = 0.05 N_0$  

\noindent ANALYSIS OF THE DATA 
%\item {}THE WHOLE RABBLE TOGETHER WITH THE MORAL SUPPORT OF THE TA 
\begin{enumerate}
\item[] \vspace*{-.08in } Plot the the natural logarithm  of the counts $N$ versus the absorber thickness $t$. Draw a straight line through the experimental points, and find the corresponding absorption length $\lambda$ by measuring the slope of the line.
\end{enumerate}
\noindent QUESTIONS    
\begin{enumerate}
\item[Q1:] \vspace*{-.08in}Does the absorption of $\gamma$ rays depend on the $Z$ of the 
absorber?
\item[Q2:] \vspace*{-.08in}Why are $\beta$ rays so easily absorbed?
\end{enumerate}
