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\begin{document}
\begin{flushright}
(revised 3/11/2014)      % 3/27/90)
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\vspace{.5in}
\begin{center}
{\bf BETA DECAY}\\
\vspace{.625in}
{Advanced Laboratory, Physics 407 } \\
{University of Wisconsin} \\
{Madison, Wisconsin 53706} \\
\vspace{.8in}
\end{center}
\begin{abstract}  
The beta decay spectrum of Cs$^{137}$ is measured using a magnetic 
spectrometer and a Geiger counter detector for the electrons. 
A Kurie plot analysis is made to extract the beta decay parameters and 
compare to theory. The internal conversion line resulting from the 
decay scheme of Cs$^{137}$ is also studied. 
\end{abstract}

\newpage
\section*{\underbar{\bf Theory}}
\smallskip
Beta decay refers to the process in which a nucleus spontaneously breaks down 
by emitting an electron (a ``Beta ray") and an antineutrino.
The process is seen in the decay of a free neutron to a proton, electron and an 
antineutrino, releasing 785 keV with a half life of 10.5 minutes:
$${\rm n}~\rightarrow~{\rm p}~+~e^-~+~\bar\nu_e$$
The process is also seen in the decay of nucleii which have too many neutrons 
to be stable.  The process always increases the atomic number Z of the nucleus 
due to the extra proton which is created.  The example used in this experiment 
is:
$$ {\rm Cs}^{137} \rightarrow {\rm Ba}^{137} + e^- + \bar\nu_e$$

The neutrino was postulated in 1931 by Pauli to explain how in beta decay the 
following could be conserved:
\begin{itemize}
\item energy -- if no neutrino, then energy disappears.
\item linear momentum -- the p and e$^-$ do not recoil in opposite directions.
\item angular momentum -- if no neutrino, then 1/2 $\hbar$ of angular momentum
disappears.
\end{itemize}

There are several features which should be noted:
\begin{enumerate}
\item The antineutrino is also required to balance the ``e-lepton number." 
It is possible to assign such a quantum number to particles; for 
example:\\[1ex]
%
%\settabs \+ nuribba & urnbg & eirnviwrengfirntiw & eubeabgiorvnwur & efb & 
%ubb \cr
\begin{tabular}{clllc}
 $e^-$ & electron & e-lepton number  & = & +1 \\
 $e^+$ & positron &                  & = & \hfil $-1$ \\
 $\nu_e$ & e-neutrino &              & = & +1 \\
 $\bar\nu_e$ & e-antineutrino &      & = & \hfil $-1$ \\
& muon, neutron, proton, $\gamma$ etc.&& = & \ 0 
\end{tabular}\strut\\[1ex]
The conservation of ``e-lepton number" expresses the fact that we have never 
observed electrons or e-neutrinos being created or destroyed except in pairs.

Although not needed in this experiment, you should know of a similar 
$\mu$-lepton number which is also conserved.
\\[1ex]
\begin{tabular}{clllc}
$\mu^-$ & + muon & $\mu$ lepton number  & = & +1 \\
 $\mu^+$ & $-\hbox{muon}$ &                  & = & \hfil $-1$ \\
 $\nu_{\mu}$ & $\mu$ neutrino &              & = & +1 \\
 $\bar\nu_{\mu}$ & $\mu$ antineutrino &      & = & \hfil $-1$ \\
& electron, neutron, proton, $\gamma$ etc. & & = & \ 0 
\end{tabular}\strut\\[1ex]
For example, the decay of a muon must obey both conservation laws (check this):
$$\mu^-~\rightarrow~e^-~+~\bar{\nu}_e~+~\nu_{\mu}$$

\item The neutrino has unusual properties:
\begin{enumerate}
\item It must move at or about the velocity of light because it has a 
very small if not exactly zero rest mass.

\item It has a very small interaction with normal matter.  A neutrino can 
easily pass through a light-day of lead.

\item It has a spin of ${1 \over 2}$ (similar to that of an electron rather than 
the spin 1 of a photon).

\item The spin of a neutrino is about the direction of motion.  The spin 
vector of a neutrino is opposite to its momentum and that of an antineutrino is 
parallel to its momentum.

\item It has zero charge.
\end{enumerate}
\item Some nucleii which are proton-rich spontaneously emit positrons by 
changing a proton to a neutron.
$${\rm p} \rightarrow~{\rm n}~+ e^+ + \nu_e$$
This occurs only within a nucleus since a proton has a smaller rest mass than a 
neutron and so cannot decay by itself.

\item Energy and momentum must be conserved in the decay.  Hence, 
depending upon the angles of the ejected particles, various energy values are 
found for the electrons.   It was this distribution of observed electron 
energies as well as the imbalance of spin angular momentum 
which led to the prediction of the neutrino as a particle with no mass 
or charge.

\item Beta Decay is an example of the weak interaction.  Remember that all 
interactions can be classified as either strong, electromagnetic, weak, or
gravitational.

\item An electron can be knocked out of the electron ``cloud" of an atom 
by a photon.  This is the Photo-Electric effect or the Compton Effect (not Beta 
Decay). If the photon originates in the nucleus and knocks out an atomic  
electron, then that phenomenon is called ``Internal Conversion".  In this 
experiment both beta decay electrons and internal conversion electrons
are detected. 
\end{enumerate}
%\newpage
%The following is extracted from ``Atomic and Nuclear Physics" by D. L. 
%Livesey (Blaisdell Publishing Co., 1966)

%\vfill
%\epsfysize=.95\textheight \epsfbox{beta_insert.eps}
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\subsubsection*{\underbar{Momentum Spectrometer}}

We first review some relativistic equations.  The relativistic momentum is 
defined $\vec p~ =~ \gamma m_o \vec v$ where $\gamma$ = (1 - ${v^2 \over c^2})^
{-1/2}$. Force is equal to the time rate of change of momentum:
\begin{eqnarray*}
\vec F = {d\vec p \over dt} &=& {d \over dt} (\gamma m_o\vec v)\\
 &=& \gamma m_o {d\vec v \over dt} ~ + ~ {d\gamma \over dt} m_o\vec v \\
 &=& \gamma m_o {d\vec v \over dt} ~ + ~ {d \over dt} (1 - {\vec v.\vec v \over 
       c^2})^{-1/2} m_o\vec v \\
 &=& \gamma m_o {d\vec v \over dt} ~ - ~ 1/2~ m_o\vec v (1 - {\vec v.\vec v \over 
       c^2})^{-3/2} (- {2 \over c^2} \vec v. {d\vec v \over dt}) \\
 &=& \gamma m_o\vec a ~+~ \gamma^3m_o\vec v ({\vec v.\vec a \over c^2}).
 \end{eqnarray*}
where $\vec a$ is the acceleration.  Check that this agrees with $\vec F = 
m\vec a$ if $v << c$.

In this magnetic spectrometer, the force F is caused by the field B:
$$\vec F = e(\vec v \times \vec B).$$
The acceleration may be found from the two equations above:
$$e(\vec v \times \vec B) = \gamma m_o\vec a ~+~ \gamma^3m_o\vec v ~~({\vec v.
\vec a \over c^2}).$$
This is a vector equation. Let $\vec a = a_\bot + a_{\parallel}$ where $a_\bot$ 
is perpendicular to $\vec v$ and a$_{\parallel}$ to $\vec v$.

The components parallel to v are:
\begin{eqnarray*}
0 &=& \gamma m_o a_{\parallel} + \gamma^3 m_o v~\big({va_{\parallel}\over
 c^2}\big) \\
&=& \gamma^3 m_o a_{\parallel}, ~~~~~ {\rm hence~}a_{\parallel} = 
0. 
\end{eqnarray*}
The components perpendicular to v are:
$$e (\vec v~ \times ~\vec B) =~ \gamma m_oa_{\bot},$$
hence $a_\bot = ({evB \over \gamma m_0})$

In this spectrometer, the field B is perpendicular to $v$.  The acceleration 
causes a circular orbit of radius r:
$$ a~=~{v^2 \over r}.$$
Hence, 
\begin{eqnarray*}
{v^2 \over r} &=& {evB \over \gamma m_o}\\
\gamma m_o v &=& Ber \\
p &=& Ber. 
\end{eqnarray*}

In this experiment, the field direction is chosen so that negative charges 
have a trajectory leading to the detector. The source emits electrons with various 
angles and various momenta.

\newpage

\begin{enumerate}
\item Variation of Angle.   Consider several electrons emitted with the 
same momenta but at slightly different angles.  Each particle will follow a 
circular path with a slightly different center point.

\includegraphics[height=4.0in]{beta_var_angle.eps}

%\epsfysize=.4\textheight \epsfbox{beta_var_angle.eps}
%\special{epsf=beta_var_angle scale=0.9}

Note that the particles will pass through a focus point one diameter away from 
the source.  Of course, if the angles are very large then the particle 
will pass near the focus.  You may like to show that the particle will 
miss the focus by a distance $\epsilon$.
$$\epsilon \simeq r~\sin^2 \theta$$
where $\theta$ is the angle shown.

\includegraphics[height=4.0in]{beta_miss_focus.eps}
%\epsfysize=.3\textheight \hspace*{.5\textwidth}\epsfbox{beta_miss_focus.eps}
%\special{epsf=beta_miss_focus scale=0.9}

Because $\sin^2\theta$ changes slowly near $\theta$ = 0, $\epsilon$ is 
very small.  A baffle with a slit is used to restrict the detected particles to 
those with $\theta$ small.  In this spectrometer the restriction is: 
$$\theta < 1/4 ~{\rm radian}~~~~~~~(14.3^{\circ})$$
$$\sin^2\theta<0.0612$$

\item Variation of Momenta.


\vspace*{2ex}
\includegraphics[height=4.0in]{beta_var_momenta.eps}
%\epsfysize=.5\textheight \epsfbox{beta_var_momenta.eps}
%\special{epsf=beta_var_momenta}


The particles of different momenta will pass through different focus points.  
By using a small source and a small detector only those in a small range from 
$p$
to $p+dp$ will be accepted.

The two effects above limit the resolution which is defined by:
\begin{eqnarray*}
R &=& {\Delta p \over p} \\
             &=& {1 \over 2r}\left[ s + w + r (\Delta\theta^2 + 
             \Delta\theta\psi^2)\right],
             \end{eqnarray*}
where: 
\begin{itemize}
\item s and w are the source and detector widths respectively.
\item $\Delta\theta = d/2r$ is the horizontal acceptance angle where d is 
the baffle opening.
\item $\psi = h/2\pi r$ is the vertical angle subtended between the source and 
detector, h being the source height.
\end{itemize}

The fractional transmission of the spectrometer is given to be
$f = {\Delta\theta\psi \over 4\pi}.$
\end{enumerate}

\subsubsection*{\underbar{Beta Decay of $\rm Cs^{137}$}}

The figure shows that two different decays are possible.  92\% of the electrons 
from Cs$^{137}$ are associated with a transition having an $E_{\rm max}$ of 514 
keV and 8\% with an $E_{\rm max}$ of 1176 keV. Some of the 662 keV Gammas are 
internally converted.

\vspace*{2ex}
\includegraphics[height=1.7in]{beta_decay.eps}
%\epsfxsize=\textwidth \epsfbox{beta_decay.eps}
%\special{epsf=beta_decay}

\subsubsection*{\underbar{Intensity}}

It is possible to calculate the probability that an emitted electron has a 
momentum between $p$ and $p + dp$.  This probability is proportional to $dp$ (if dp 
is small enough) and is a function of $p$.  Therefore, we label this probability: 
$I(p)dp$.

The theory for Beta Decay was developed by Enrico Fermi in 1934.  This theory 
assumes that the neutrino has zero mass and predicts:

$$I(p)dp = C F(Z,p)~(E_{\rm max} - E)^2~ p^2~ dp$$

\noindent
where:
\begin{itemize}
\item $I(p)dp$ is the probability of a momentum between p and p + dp

\item $C$ is a constant for a particular beta decay transition.

\item $F(Z,p)$, the ``Coulomb factor", takes account of the effect of the 
nuclear electrostatic field.  It may be found in tables in the references and 
for energetic betas $F\approx 1$.

\item $Z$ is the atomic number of the nucleus.

\item $E_{\rm max}$ is the maximum energy of the spectrum considered, 
i.e. the total kinetic energy of the beta plus that of the neutrino.

\item E is the kinetic energy of the beta particle which has momentum p. Since 
the energies involved are of the order of the rest energy $m_0c^2$, the correct 
expression to use is:
$$E = (m_0^2c^4 + p^2c^2)^{1/2} - m_0c^2$$
where $m_0$ is the rest mass of the electron and $c$ is the velocity of 
light.
\end{itemize}

A convenient way of plotting experimental results for comparison with the Fermi 
theory was suggested by Kurie in 1936 and is called a Kurie plot.

\vspace*{2ex}
\noindent
Define y:
$$y^2 = {I(p) \over Cp^2F(Z,p)},$$
then ~~~$y = E_{\rm max} - E$.

\vspace*{2ex}
A plot of $y$ against $(x = E)$ will give a straight line with negative slope 
which intercepts the $x$ axis at $E_{\rm max}$.

\subsubsection*{\underbar{The x Axis of the Kurie Plot}}

An expression for the energy as a function of the magnetic field is needed.  By 
means of the previous calibration of the field, the energy may be expressed as 
a function of the current.
\begin{itemize}
\item The rest energy is $E_0 = m_0c^2$.

\item The total energy is $E_T =\sqrt{E_0^2 + p^2 c^2}~~~
({\rm also}~ E_T = \gamma 
m_0c^2)$.

\item The kinetic energy is $E = E_T - E_0$
\begin{eqnarray*}
E  &=&  E_T - E_0\\
&=& \sqrt{E_0^2 + p^2 c^2} - E_0 \\
                &=& \sqrt{E_0^2 + (Berc)^2} - E_0.
                \end{eqnarray*}

\item If the energies are expressed in electron volt units:
\begin{eqnarray*}
E &=& \sqrt{E_0^2 + (Brc)^2} - E_0  \\
{\rm where}~ E_0 &= 511003.4 \pm 1.4~ \rm eV.
\end{eqnarray*}
\end{itemize}
From each magnet current I we can obtain B in Tesla(webers/meter$^2$) 
and then by 
using $r$ and $c$ (in mks units) we obtain the kinetic energy $E$ for the $x$ 
axis.

\subsubsection*{\underbar{The y Axis of the Kurie Plot}}

If the counting rate is $A$, the range of momentum $dp$, $I$ the intensity, and 
$\Delta p/p$ the momentum acceptance of the spectrometer, then:
\begin{eqnarray*}
dA &=& I(p) ~dp \\
           A &\approx& I(p)p(\Delta p/p). 
\end{eqnarray*}
The y axis variable for the plot is:
\begin{eqnarray*}
y &=& \sqrt{I(p) \over CF(Z,p) ~p^2}\\
             &=& \sqrt{A \over CF(z,p) p^3 (\Delta p/p)}\\
             &=& \sqrt{A \over CF(z,p) B^3 e^3 r^3 (\Delta p/p)}.
\end{eqnarray*}
The momentum acceptance $(\Delta p/p)$ 
is a geometrical factor and so is a constant for a particular spectrometer. 
Also $CF(z,p)\simeq 1$.

We need not worry about the value of the constants because we want the 
intercept on the $x$ axis and so will use:

$$y = \sqrt{{A \over B^3}}.$$


\section*{\underbar{\bf Equipment}}

\subsubsection*{\underbar{Spectrometer}}

The Atomic Laboratories Beta Ray Spectrometer deflects an electron through an 
angle of 180$^\circ$ between two points (source and detector) spaced a distance 
2r apart (r is nominally 1.5 in).  The 180$^\circ$ point on the trajectory is 
called the ``first-order focus", because electrons emitted with small angular 
deviations from the central trajectory tend to be focussed at 180$^\circ$.

The principle of the instrument is discussed in the theory section.  The 
dimensions are:
\begin{center}
\begin{tabular}{lccl}
{\rm source width}   &$s$ &$\simeq {1 \over 8}$ {\rm in} \\
{\rm detector width} &$w$ &= ${1 \over 16}$ {\rm in} \\
{\rm radius}         &$r$ &= 1.5 {\rm in} \\
{\rm gap in baffle}  &$d$ &= 0.75 {\rm in} \\
{\rm source height}  &$h$ &$\simeq {1 \over 8}$ {\rm in} 
\end{tabular}\end{center}

From these parameters, the momentum acceptance $\Delta p/p$ may be calculated:
$$\Delta p/p = 0.10\ (10\%).$$

The fractional transmission which determines the counting rate is:
$$f = 1.05 \times 10^{-3}.$$

In order to use the spectrometer with weak sources, which do not require 
special safety rules, 
it was necessary to accept a relatively poor resolution so that the 
transmission would not be too low.  Obviously a stronger source would allow us 
to improve the resolution by decreasing $s$, $w$ and $d$.

\subsubsection*{\underbar{Source}}

Students should \underbar{not} normally 
remove the source which is on the upper plug of 
the spectrometer.  The $\rm Cs^{137}$ source (approximately $10~\mu {\rm Ci}$) 
is mounted 
on a thin aluminized mylar film, which in turn is mounted on a small brass 
ring.  The ring and the source are removable and are held in place by an 
internal spring clip.  Simply grasp the ring firmly and pull it away after the 
source holder has been removed from the Beta Ray Spectrometer. (Cenco sells an 
optional Sodium-22 source on the same size ring and this ring in turn can be 
mounted on the same source holder merely by pushing it in over the spring 
clips.) During insertion or removal of the source, care has to be taken 
not to touch 
the source itself which is in the center of the mylar film.  It is unprotected 
and will scrape off at the least touch.  If it were protected, it would absorb 
so many beta rays that the experiment could not be conducted properly.

\vspace*{2ex}
\noindent
CAUTION: WHEN INSERTING OR REMOVING THE CESIUM, SODIUM, OR ANY OTHER 
RADIOACTIVE SOURCE, GLOVES SHOULD BE WORN AND HANDS SHOULD BE WASHED BOTH
BEFORE AND AFTER THE OPERATION. IT IS QUITE POSSIBLE THAT THE SOURCE, IF 
TOUCHED DIRECTLY, WILL COME OFF ON THE HAND.  GREAT CARE SHOULD BE EXERCISED TO 
SEE THAT STUDENTS OR UNQUALIFIED PERSONNEL ARE NOT ALLOWED TO TOUCH THESE 
SOURCES DIRECTLY.

\subsubsection*{\underbar{Baffle}}

This is mounted inside the center plug.  Students should not remove the baffle 
since it can easily be damaged.  It is a horizontal plate (1/16 in thick) 
with a rectangular hole.

\subsubsection*{\underbar{Detector}}

This is mounted on the lower plug.  It is an end-window Geiger counter.  You 
may dismount it to inspect it but you should be very careful.  The Tube 
has a mica end window with a thickness of only 1.4 - 2.0 mg/cm$^2$ and is quite 
fragile.  It should not be touched and the tube should be capped when not in use.  
The tube should not be used above 700 volts.

\def\ll{\leftline} \def\ub{\underbar}

\subsubsection*{\ub{Probe Port}}

A Hall probe may be inserted near the baffle by removing the baffle plug.
The magnetic field must be measured with no vacuum in the system. The field
should be unchanged when the chamber is evacuated. 

\subsubsection*{\ub{Vacuum}}

The electrons are far more energetic (up to 660 keV) and so do not need as good 
a vacuum as electronic vacuum
tubes (0.1 to 0.3 keV) and television tubes (up to 21 
keV). The system will 
operate with a pressure as high as 1 mm of Mercury (1 Torr) but better 
results are obtained if the pressure is kept less than 100 microns (10$^{-1}$ 
Torr).  The normal vacuum pump on the apparatus will maintain 10 to 20 microns 
(10$^{-2}$ to 2 $\times 10^{-2}$ Torr).
A thermocouple gauge is used to measure the pressure.

\subsubsection*{\ub{Detector High Voltage}}

A positive regulated supply set to about $+500$ V is connected to the Geiger
counter via a RC decoupling network. The pulse output from the Geiger counter
is capacitively decoupled from the Geiger counter HV and is connected from the
decoupling network box to the preamplifier input.

\subsubsection*{\ub{Preamplifier}}

ORTEC Model 435A. Should be set for NEG input pulses and UNIPOLAR output
pulses. The gain should be set to drive the ORTEC scaler. A setting of coarse
gain = 16 should be about right.

\subsubsection*{\ub{Scalar}}

ORTEC Model 484. The Scaler records single counts directly, provided that the
input pulses (positive) are above the threshold setting. You will have to time
the runs. A clock is provided on the bench.

\subsubsection*{\ub{Magnet}}

This is a simple electromagnet which has 2 coils connected in series.  The 
current is supplied by a variable regulated voltage source (0 to 50 VDC) and is 
monitored by a series ammeter.

\section*{\ub{\bf{Procedure}}}
\begin{enumerate}
\item  Read the theory and equipment sections.

\item Predict the direction of the magnetic field to bend electrons from the 
source to the detector and connect the power leads accordingly.
Set the current to about 0.25 amp.

\item Calibrate the Hall Probe by using the internal calibration adjustment. 
(see Instructor) 
Insert the Hall probe into the probe port and plot the field B in Tesla
against the current I.  Take care not to strain the probe---clamp it in 
position---don't let it hang by its delicate probe.  Remember to rotate the 
probe to 
find the maximum reading and also to allow for magnet hysteresis 
by always increasing I towards its measured value.  (1 Tesla = 10,000 Gauss)

\item Pump down the spectrometer.  The pressure should be less than 100 
microns after a minute.  Check that no radioactive sources (from adjacent 
experiments) are nearby.

\item Set the current to about 0.25 Amp and measure the counting rate for 
various high voltages from 400 volts up to 700 volts (but not beyond).  This 
data should show a plateau.  Select a high voltage at the center of the plateau 
and use it for all following measurements.

\item Measure the counting rate A(I) for various currents from 0 up 
to 0.90 Amps.  Remember that the standard deviation 
expected on $N$ counts is $\sqrt{N}$.

\item Plot the counting rates A(I) against the current I.

\item Identify the critical regions of $E_{\rm max}$ and the conversion line.
Take additional measurements in these regions in order to define them 
more accurately.   

\item Measure the counting rate at 0.9 amps carefully, regard it as a background 
and subtract it from the other rates.  

\item Assume the Hall Probe is linear, $c=2.9979\times 10^8$ m/sec and use the 
given value of $r$.  Calculate $pc/e = Brc$ for each current I. 

\item Calculate the energy (in keV) of the internal conversion peak from the  
observed momentum $p$ by using the value of $Brc$ for the peak and the electron
rest mass $E_0$ = 511003 eV.  Then calculate:
$$E = \sqrt{E_0^2 + (Brc)^2} - E_0.$$
How does your result compare with the accepted value? 

\item Calculate the x axis and y axis points of the Kurie plot for each 
current I.

\item Tabulate I, A, B, Brc, the x value, and the y value. 

\item Make the  Kurie plot.  Draw a straight line on the part expected to be 
linear and determine $E_{\rm max}$ of the Beta Decay. Compare your result 
with the accepted value. 

\item You may want to make another Kurie plot for the data at energies 
greater than internal conversion peak. This plot should give an E$_{\rm max}$
of 1176 keV. 
\end{enumerate}
\end{document}