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(revised 10/20/10)
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\begin{center}
{\bf{\uppercase{Range of Alphas}}}\\
\vspace{.625in}
{Advanced Laboratory, Physics 407 } \\
{University of Wisconsin} \\
{Madison, Wisconsin 53706} \\
\vspace{.8in}
\end{center} 
\begin{abstract}
A silicon solid state detector is used to measure the energy of alphas 
which have passed through air from an Am$^{241}$ source. The 
air pressure may be 
varied and so the Bethe-Bloch formula
for $dE/dx$ may be verified and the alpha particle range 
and straggling can be measured. 

\end{abstract}

\newpage
\section*{\bf\underbar{Theory}}

A charged particle moving through a neutral medium will interact 
electromagnetically with both the electrons and nucleii of the material. The 
electromagnetic interactions with the nucleii cause Rutherford Scattering and 
are seen as small (and occasionally large) changes in directions. The 
interactions with the electrons are far more frequent and are seen as a 
fairly steady loss of kinetic energy.

There are, of course, statistical fluctuations in the rate of the
interactions and this is seen as ``straggling" of the range of monoenergetic
particles. For example, alphas with a mean range of 20 cm will have range
fluctuations (straggling) of about $\pm 1$\%. 

The rate of loss of energy can be calculated and is expressed by the 
Bethe-Bloch formula.(see Ref.\cite{evans}, pg. 637
and Ref.\cite{melissinos}, pp. 155-162.)

\begin{equation}
{dE \over dx} = - { 1 \over (4 \pi \epsilon_o)^2} { 4 \pi e^4 z^2 NB \over 
m_e v^2}\ \ \ (mks\ units) 
\end{equation}
where 
\begin{eqnarray*}
e &=& \mbox{charge on electron (coulombs).}\\
z &=& \mbox{Atomic number of moving particle.}\\ 
N &=& \mbox{the number of atoms/unit volume (meter$^{-3}$).} \\
m_e &=& \mbox{mass of electron (kg).} \\
v &=& \mbox{velocity of the moving particle (meter/sec).} \\
E &=& \mbox{kinetic energy of the moving particle (joules).}\\ 
x &=& \mbox{distance travelled by the particle (meter).} \\
\epsilon_o &=& \mbox{permittivity of free space.} \\
1/(4 \pi \epsilon_0) &=& \mbox{$8.988 \times 10^9$ Newton
meter$^2$/coulomb$^2$. }\\
B &=& \mbox{Atomic stopping number (dimensionless).}\\
\end{eqnarray*}

The factor $B$ is not constant but varies slowly with energy in a
logarithmic manner. The theoretical calculations for $B$ become difficult when
allowance is made for the partial screening of the nuclear charge by the inner
(K) electrons.
\newpage


The best formula for $B$ is probably that of Ref.\cite{evans}, pg. 638. 
\begin{equation}
B=Z\left[\ln\Bigl({2m_ev^2\over <I>}\Bigr)-\ln(1-\beta^2)-\beta^2\right]-C_K
\end{equation}
where
\begin{eqnarray*}
C_K &= &\parbox[t]{.8\textwidth}{the correction term for the K-shielding. The
equations and a plot of $C_K$ are shown on pg. 639 of Ref.\cite{evans}}\\
\beta & = &\mbox{ the usual relativity factor - veloc 
particle/veloc light}\\
<I> &= &\mbox{``average" ionization potential of the stopping medium}\\
Z &= &\mbox{ (average) atomic number of the stopping medium}
\end{eqnarray*}

If we assume that the velocity is non-relativistic, then $E={1\over2}mv^2$ and
so:
\begin{equation}
{dE\over dx}=-{1\over(4\pi\epsilon_0)^2}2\pi e^4z^2N\left({M\over m_e}\right)
{B\over E}
\end{equation}
where $M$ is the mass of the particle.\\
$\beta^2$ is also small so that $\ln(1-\beta^2)\approx-\beta^2$ and so
\begin{equation}
B=Z\ln\left({4m_eE\over M<I>}\right)-C_K.
\end{equation}
Fortunately in our experiment with alpha particles in air, the calculated value
of $C_K$ is nearly constant near 0.90.

Numerical values for $\frac{dE}{dx}$ as a function of energy are shown in Fig. \ref{dedx} 
for various particles. A 5 Mev alpha particle has a $\frac{dE}{dx}$ value of about 
$1000\ {\rm MeV/gm/cm^{-2}}$. $\frac{dE}{dx}$ has minimum for all particles at a kinetic emergy
of about twice the rest mass. For singly charged particles this value is about 
$2\ {\rm MeV/gm/cm^{-2}}$. 
\begin{figure}
\begin{center}
\includegraphics[width=5.0in]{dEdx.pdf}
\caption{\label{dedx}Energy Loss Curves for Different Particles in Air and Lead.}
\end{center}
\end{figure}
The derivation of the Bethe-Bloch equation assumes that the projectile velocity 
is much larger than the characteristic orbital electron velocities. When the projectile
velocity is lower than the requirements of this condition,
dE/dx must go to zero as the particle energy approaches zero. A plot of the energy 
loss of the ionizing radiation as the particles travel through 
matter is called a Bragg plot. The Bragg plot will show a peak just before the 
particles come to rest. Fig.2 shows a Bragg plot for 5.49 MeV Alpha particles in air
at STP. The peak in this case corresponds to an Alpha particle energy of about 1 MeV.
\begin{figure}
\begin{center}
\includegraphics[width=4.0in]{BraggCurveforAlphasinAir.pdf}
\caption{Bragg Curve for 5.49 MeV Alpha Particles in air at STP}
\end{center}
\end{figure}

\vspace{0.25in}
The particle range can be determined by integrating $\frac{dE}{dx}$ over the particle
energy.
\begin{equation}
R(E)=\int\limits_{0}^{E}\frac{dE^{\prime}}{-\frac{dE^{\prime}}{dx}}.
\end{equation}
Statistical fluctuations lead to a distribution of ranges about the mean $R_0$ with 
a straggling parameter $\alpha$ defined by the probability distribution for the ranges:
\begin{equation}
P(R) = \frac{1}{\alpha\sqrt{\pi}}\ {\rm exp}[-\frac{{(R-R_0)}^2}{\alpha^2}].
\end{equation}


Fig. \ref{range} shows differential and integral probability distributions for heavy particles
ranges. The extrapolated range $R_{ex}$ is related to the straggling parameter $\alpha$ as 
explained later in the text.  

\begin{figure}
\begin{center}
\includegraphics[width=5.0in]{alpharange.pdf}
\caption{\label{range}The differential and integral probability distributions
for heavy charged particle range.}
\end{center}
\end{figure}

Fig. \ref{air} shows range vs energy for alpha particled in air at standard conditions. The range for
a 5486 keV alpha in dry air at $15^{\circ}$ and 760 Torr is 4.051 cm. 

\begin{figure}
\begin{center}
\includegraphics[width=5.0in]{air.pdf}
\caption{\label{air}Alpha Range-Energy in air}
\end{center}
\end{figure}

\section*{\ub{\bf{Apparatus}}}
\begin{enumerate}
\item\ub{Vacuum Chamber.} The vacuum chamber is made of pyrex glass and is 
mounted inside a dark wooden box. The Silicon 
detector must be protected from room light since photons cause a noise
background which spoils the resolution of the detector. 
\underbar{Do not open the
glassware}. The alpha source could, over a long time, shed some radioactive
dust. The system has been designed to ensure that any dust is trapped inside
the glass system vacuum. 
The source to detector surface was remeasured in 2005 and determined to be 
$6.65\pm 0.05$ cm.
\item \ub{Pressure Gauge}---MKS Baratron type 122A. The pressure gauge is
an absolute pressure transducer based on measuring the capacitance of a
sample chamber. The accuracy is rated at 0.5\% of reading from 
$0-50^\circ$C. The readout is in Torr.
\item \ub{Vacuum Pump.} The pump is a 2 stage rotary pump enclosed in the
standard cart to reduce acoustic noise. Notice that a flow of warm air is
exhausted from the cart by an electric fan to prevent the pump and motor from
overheating. 
\item \ub{Alpha Source.}
\begin{figure}
\begin{center}
\includegraphics[width=4.0in]{alpha_source.pdf}
\caption{\label{amdecay}Decay Scheme of Am$^{241}$}
\end{center}
\end{figure} 
The decay scheme of Am$^{241}$ is shown in Fig \ref{amdecay}
This source is not sealed and so must remain inside the vacuum 
enclosure. (Usually sources are sealed with very thin metal skins. In this 
experiment, a skin would slow the alphas slightly. As the skin could not have 
a perfectly uniform thickness, the alphas would emerge with a broader range of 
energies.) The 5486 and 5443 keV alphas will not be resolved due to the finite 
thickness of the source. A 5486 keV alpha 
has a range in dry air at 15$^\circ$C and 760 Torr of 
4.051 cm and so the source alphas cannot reach the detector until the chamber 
pressure has been reduced. 

\item \ub{Solid State Detector} - (Ortec A-040-200-300, Serial 9-129B).
The detector is a surface barrier detector consisting of an extremely thin
p-type layer on the face of a high purity n-type Si wafer. The rated energy
resolution of the detector is 40 keV and the active thickness when fully
depleted is $300\ \mu$. The p-type surface of the detector is gold plated 
with a layer approximately 40 $\mu$g-cm$^{-2}$ thick. The detector has
a sensitive diameter of about 16 mm and is mounted on a BNC connector
within the vacuum system. Although the detector can operate with a bias of
+100V in a very good vacuum, we will use the detector in the dangerous
$10^{-2}$ Torr to 10 Torr region. 
Set the bias to 30 V but do not use a bias greater than +30 V.
Some useful properties of Si are listed in Fig. \ref{Siprop}.

\item \ub{Pre-amplifier} - (Ortec model A576). This is a charge sensitive
pre-amplifier which also supplies the bias voltage for the Si detector.
The pre-amplifier is designed to have a large effective input 
capacitance $C_a$ so that most of the charge drains from the detector and 
cable into the pre-amplifier and is amplified. If the capacitances of the 
detector and cable are $C_d$ and $C_c$, then:
$$Q_a=Q_{\rm tot}\left({C_a\over C_d+C_c+C_a}\right)\ ,$$
where $Q_{\rm tot}$ is the total charge collected by the detector, and $Q_a$ is
the charge delivered to the charge sensitive pre-amplifier.

Although $C_a$ is large, the charge seen by the \hfuzz=8.46006pt
pre-amplifier depends upon $C_c$ and so the same short cable should be used to
connect the detector and pre-amplifier for all measurements. 
\item \ub{Amplifier.} ORTEC Model 570. Use the input set to POS and the 
unipolar output.  The amplifier gain is adjustable so that the gain can be 
matched to
the full scale range of the PC MCA System. The amplifier also shortens the
pulses so that a typical alpha pulse out is $\sim 2\ \mu sec$. 
\item \ub{Pulse-Height Analyzer} - The PHA (Spectrum Techniques UCS30) is
an external  circuit component with a USB connection to the computer. The full scale
voltage is 8 V and the circuit is usually set for 512 channels full scale. 
\item \ub{Scaler} - (Ortec Model 484).
\end{enumerate}
\begin{figure}
\begin{center}
\includegraphics[width=4.0in]{alpha_last.pdf}
\caption{\label{schematic}Apparatus Schematic Diagram}
\end{center}
\end{figure}
%\newpage

\section*{\bf\ub{Procedure}}
\begin{enumerate}
\item 
Read Ref.\cite{melissinos}, pp. 208-217, and the theory in Ref.\cite{evans}, 
Chapt. 22. 
\item 
Pump down the vacuum chamber. The filter has a low 
pumping speed and so the time to reach 0 Torr is several minutes.
Practice 
using the air inlet valve or the vacuum pump valve to obtain and hold any 
pressure you wish.
\item 
Connect the detector to the FET pre-amplifier with a short (1 foot) 
cable. The TEST-OFF switch should be set to the center position.
Connect the rear pre-amplifier output to 
the input of the pulse amplifier. Connect the 
pulse amplifier output to the 
pulse height analyzer input and scope. Do not terminate the cable to the scope, 
since the pulse height analyzer input has a relatively low input 
impedance. The schematic is shown in Fig. \ref{schematic}. 
\item
The amplifier should be set for POS input pulses.
Lower the pressure to less than 5 Torr and look 
for positive pulses $(5\ \mu$sec) at the PHA and scope inputs. 
Adjust the gain of the amplifier so that the pulses are being 
counted near the upper end of the 512 channel PHA. 
The PHA requires positive pulses and 
full scale corresponds to 8 Volt input pulses.
Record all parameters so that you can later reproduce the 
same gain. 
The Si detector output will be pulses whose amplitudes are proportional
to the alpha particle energy less the energy lost in the air.  
Since these pulses are fed to the PHA we have:
$$\hbox{Energy}=\hbox{constant}\times\hbox{pulse
height}=\hbox{constant}\times(\hbox{channel number}+\hbox{constant})$$
\item 
The bias voltage required to fully deplete the Si detector is 30 V.
Check to see that the voltage is correct.
\item
Measure the full-width half maximum resolution of the alpha particle peak
and compare to the intrinsic resolution of the Si detector. 
\item 
Measure the range vs pressure using the scaler. In principle, the PHA could
be used by integrating the number of counts in the alpha peak as a function of
pressure, but the pulse heights are too small near the end of the range and the
dead time correction will be very large due to the high alpha particle counting rates.  
To configure the scaler-amplifier system,
set the amplifier gain to the maximum value and set the pressure above the
alpha particle range. Lower the amplifier gain so that the noise 
pulses are barely counted by the scaler. 
Measure the count rate as a function of pressure starting at the high 
pressure end. The pressure steps will have to be very small near the end of the
range to accurately measure the range and straggling parameter. 

Plot count rate against P. Determine the mean range $R_0$ and 
the extrapolated range $R_{\rm ex}$ of the alpha particles as shown in Fig.\ \ref{range}.
Compare your result to the predicted value based on Fig.\ \ref{air}.
Remember that you have to correct the predicted range to the current air temperature.
From the quantity $R_{\rm ex}-R_0$ determine the \underline{effective}
straggling parameter
$\alpha$. The quantity $\alpha$ is defined by the probability 
distribution for the ranges:
$$P(R) = \frac{1}{\alpha\sqrt{\pi}}\ {\rm exp}[-\frac{{(R-R_0)}^2}{\alpha^2}]$$
and $R_{\rm ex} - R_0 = \frac{\sqrt{\pi}}{2}\alpha.$
Check that the source-detector distance is consistent with the value 
given earlier in the writeup. Compare the 
measured straggling parameter to the value given in Fig. \ref{straggle} below. 
\item 
Now use the PHA to measure the alpha particle energy as a function of air 
pressure. Start at 0 Torr (or as far as the system will pump down) and record the 
channel peak as a function of pressure in about 25 Torr steps. The channel 
number at 0 Torr will correspond to the alpha particle energy of 5.49 MeV.
Record the channel peak down to the highest pressure at which you can 
still determine the peak channel. Plot E (channel number) as a function of pressure. 
This is the primary data from which you will determine dE/dx by numerically
evaluating the derivative from the E vs pressure curve.

Evaluating the derivative of the E vs P curve can be done numerically by taking
the differences between successive E (channel number) points and dividing by the 
pressure difference. Plot dE/dx vs P and dE/dx vs E. Note that dE/dx is energy
dependent and rises as the energy decreases. The energy dependence would go
as 1/E if not for the ln E dependence. 

\item Plot $E \left({dE \over dx} \right)$ vs $ln E$. Since there is no
additional E dependence, this procedure should result in a straight line
from which the average ionization potential of air, $<I>$, can be determined.
This can be readily done by extrapolating the straight line to zero and
finding the intercept on the ln E axis. Determine this value of E
and use it to solve for $<I>$ in eV. Compare to the expected value of
about 100 eV. You will have to provide a value for $m_e\over M$. 

\item 
Both the source and the detector have finite widths and so some 
particles will travel slightly different path lengths to the detector.
Discuss this contribution to the observed energy resolution. 
\item From the count rate and by estimating the source and detector 
dimensions (without opening the chamber), estimate the source strength in 
microcuries ($\mu$Ci).
\item 
Use the range-energy data sheet in Fig. \ref{silicon} to check that 
the solid state detector 
has a depletion depth greater than the range of 5.5 MeV alphas.
\end{enumerate}

%\section*{\bf\ub{References}}
\begin{thebibliography}{99}
\bibitem{evans} R.D. Evans, ``The Atomic Nucleus," McGraw--Hill, 1955.
\bibitem{melissinos} A.C. Melissinos, ``Experiments in Modern Physics'', 
Academic Press, 1966 (2nd Ed. 2003).
\end{thebibliography}





\begin{figure}
\begin{center}
\includegraphics[width=4.0in]{silicon.pdf}
\caption{\label{Siprop}Silicon Properties}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=5.0in]{straggle030.pdf}
\vspace{-1.5in}
\caption{\label{straggle}Alpha Particle Straggling Parameter in Air}
\end{center}
\end{figure}
\vspace{0.25in}

\begin{figure}
\begin{center}
\includegraphics[width=5.0in]{range.pdf}
\caption{\label{silicon}Alpha Range-Energy in Silicon}
\end{center}
\end{figure}


\end{document}


















