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\begin{flushright}
(revised 1/28/99)   
\end{flushright}
\vspace*{2cm}
\begin{center}
\bf STERN-GERLACH
\end{center}
\vspace*{1cm}
\begin{center}
Advanced Laboratory, Physics 407,\\
University of Wisconsin\\
Madison, Wisconsin 53706
\end{center}
\vfill
\begin{center}
\bf Abstract
\end{center}

The experiment performed by Otto Stern and Walther Gerlach in 1922 provided 
very convincing evidence of two important consequences of Modern Quantum 
Mechanics
\begin{enumerate}
\item Space quantization can occur even in non-periodic systems.
\item That some particles have an intrinsic angular 
momentum, and therefore magnetic moment.
\end{enumerate}
The experiment did not really produce any new ideas but dramatically confirmed 
ideas developed indirectly from Spectroscopy and the Zeeman Effect.  The Stern 
Gerlach experiment was in principle simple and its results were clear.  It 
removed many of the lingering doubts that Quantum Mechanics is true.

\newpage
\begin{flushleft}
\underline{\bf The Original Experiment}
\end{flushleft}

Stern and Gerlach generated a beam of neutral silver atoms by evaporating 
silver from an oven.  The process was performed in a vacuum so that the silver 
atoms moved without scattering.  The atoms were collimated by slits and sent 
through a region with a large non-uniform magnetic field.

A magnetic field non-uniformity ${\partial B_z \over \partial 
z}$ produces a force ${\partial B_z \over \partial z}\mu_z$ on a magnetic
moment
where $\mu_z$ is the component of the magnetic moment $\mu$ in the $z$ 
direction.

The silver atoms were thus deflected and allowed to strike a cold metallic 
plate.  After about 8-10 hours the number of condensed silver atoms was large 
enough to show a visible trace.  The trace showed 2 marks showing that the 
silver atoms had 2 possible components of $\mu_z$. 

This would not have been expected with classical physics since this would have 
predicted that the $z$ component of $\mu$ would have been
\[ \mu_z = \mu\cos\theta \]
where $\cos\theta$ could have all values from $-1.0$ to +1.0.

Even the original Schr\"odinger theory predicts an odd number of possible 
states.  This could explain $\mu_z$ having, for example, 3 values: $-\mu$, 
0 and
$+\mu$

The obvious 2 states shown by the experiment is evidence that something is 
missing in the original Schr\"odinger Theory.  The missing idea is that 
electrons have an intrinsic spin (a spin which cannot be removed) and that the
angular momentum can be written in the form:
\[\vec S={1\over2}\hbar\vec\sigma.\]
The component of $\vec S$ in any specified direction, say the ``$z$" or third
direction must then have eigenvalues $\pm{1\over2}\hbar$.

\begin{flushleft}
\underline{\bf This Experiment}
\end{flushleft}

We include two improvements to the original experiment  that allow it to 
be done more easily.
\begin{enumerate}
\item
Potassium is used instead of silver because:
\begin{enumerate}
\item
it is easier to evaporate (63.6$^{\circ}$C for K instead of 
961.9$^{\circ}$C for Ag)
\item
The low temperature means that the Potassium atoms are moving at a 
lower speed and are more easily deflected.
\item
Potassium has a single valence electron outside its closed shells 
and the magnetic moment is more obviously due to the single valence electron.
\item
Potassium is easier to ionize than silver and is thus easier to 
detect electrically.
\end{enumerate}
\item
A hot wire detector is used instead of a cold metallic plate.
The hot wire ionizes a fraction of the potassium atoms which strike it.
The positive recoiling ions are then collected on a nearby negative electrode 
and the small ionization current is measured by an electrometer.  This allows 
modern electronics to be used with its high sensitivity for small currents and 
fast reaction time.
\end{enumerate}
\medskip
\begin{flushleft}
\underline{\bf Theory}
\end{flushleft}

One result of Schr\"odinger's Quantum Theory is that a system existing in a 
region of space with a unique axis of symmetry (such as the $z$ 
axis) will have a 
wavefunction which can be expressed in the form of a product
\[ \psi(r, \theta, \phi) = \Xi (r, \theta) \Phi(\phi)\]

The angle $\phi$ is the azimuthal
angle measured by rotating around the axis of symmetry. 
The $r$, $\theta$ and $\phi$ form a set of spherical coordinates.  Read Eisberg 
and Resnick pages 256--259 for the solution of the equation.

The function $\Phi(\phi)$ must repeat after $\phi$ is increased by 2$\pi$ and 
consequently $\Phi(\phi)$ has a solution:
\[\Phi = e^{im\phi}\]
where $m$ is an integer. The state has an angular momentum $J_z$ about the 
$z$ axis 
which is:
\[{J_z = m\hbar}.\]

The number of such spatial states is always {\bf odd}, since $m$ is any integer 
from $-j$ to $+j$ where $j$ depends upon the form $\Xi(r,\theta)$.  
In our case of 
potassium, $j = 0$ and so only one state ($m=0$) is possible.

However the intrinsic spin (introduced by the Dirac improvement to 
Schr\"odinger's Theory) may also be included as an extra product term if the 
space has a unique axis of symmetry.  The total angular momentum about the $z$ 
axis then includes that of the intrinsic spin $\vec S$
$$ |S_z|=\textstyle{{1 \over 2}} \hbar.$$

In our case of potassium, the intrinsic spin causes the single $m=0$ angular 
momentum to be split into {\bf two} states with angular momenta of $\pm{1 \over 
2}\hbar$.  The regular Schr\"odinger Theory can only predict an {\bf odd} number 
of states and so the number of states is an important test for the existence of 
intrinsic spin.

The spin is measured by 
an effect on the associated magnetic moment. The intrinsic 
magnetic moment predicted for an electron has two possible $z$ components:
\begin{eqnarray*}
\mu_z &=& \pm {1 \over 2} {\rm (1~ Bohr~ Magneton)} \\
               &=& \pm {1 \over 2} \Bigg({e\hbar  \over 2m_e}\Bigg) \\
               &=& \pm {1 \over 2} 
(0.927 \times 10^{-23}~\hbox{Amp-meter}^2)
\end{eqnarray*}

The suggestion of intrinsic spin by Goudsmit and Uhlenbeck and the original
Stern-Gerlach experiment are discussed in Eisberg and Resnick, pages 296--302.
\par
\medskip
\begin{flushleft}
\underline{\bf The Atoms}
\end{flushleft}

Potassium atoms have 19 protons and either 10 or 12 neutrons.  The electrons 
are: \\
\rule{0pt}{20pt}\begin{tabular}[t]{lll}
$n = 0$ shell       &                     & filled with 2 \\
 $n = 1$ shell       &                     & filled with 8 \\
 $n = 3$             & $\ell$ = 0 subshell & filled with 2 \\
                   & $\ell$ = 1 subshell & filled with 6 \\
                   & $\ell$ = 2 subshell & empty \\
 $n = 4$             & $\ell$ = 0 subshell & - - one valence electron 
\end{tabular}\\[5pt]
thus 18 of the electrons form a closed core with the same configuration as in 
the Argon atom.  The 19th electron sees a potential very similar to that of a 
single proton and thus:
\begin{enumerate}
\item
The energy levels of the 19th electron (if excited above n = 4, 
$\ell$ = 0) are somewhat similar to those of a Hydrogen atom.
\item
The magnetic moments of the electrons in the core cancel to give no 
contribution to the total magnetic moment of the atom.  The nuclear magnetic 
moments are 0.391 and 0.215 nuclear magnetons for $_{19}{\rm K}^{39}$ and 
$_{19}{\rm K}^{41}$ and since a nuclear magneton is about ${1 \over 2000}$ of a Bohr 
magneton, the only significant magnetic moment is that of the valence electron.
\item
The valence electron is in an $\ell = 0$ state and so it has no 
spatial {\bf (orbital)} magnetic moment.  The atom therefore has a magnetic moment 
approximately equal to the intrinsic magnetic moment of an electron.
\end{enumerate}

The above was strongly suspected from spectroscopic studies.  The idea of an 
ad-hoc ``intrinsic magnetic moment" is however rather unsatisfactory.  The 
Stern Gerlach experiment provided a direct method
for finding the number of states 
and measuring the intrinsic magnetic 
moment.
The observation of an even number of states then confirms the concept of
``intrinsic" spin.\par
\medskip
\begin{flushleft}
\underline{\bf Magnetic Field}
\end{flushleft}

The experiment sends a beam of potassium atoms in the $x$ direction 
through a strong magnetic field gradient.  The magnetic field causes a force
\begin{eqnarray*}
\vec F &=& -\vec\nabla ({\rm potential~ of~ mag.~ mom.~ in~ field ~B})\\
            &=& -\vec\nabla (\vec\mu \cdot \vec B) 
\end{eqnarray*}
If the component of $\mu$ in the direction of $B$ is $\mu_B$, then
\[F = -\vec\nabla (\mu_BB).\]
Here we use the idea that $\mu_B$ is quantized (since $B$ provides an axis of 
symmetry) and so $\mu_B$ is independent of x, y, and z.
\[F = -\mu_B \vec \nabla B\]

The magnetic poles are designed so that there is a region in which $B$ is in 
the horizontal $z$ direction and $\vec \nabla B$ is both large and 
approximately constant.  Hence the force is simply:
\[\vec F = -\mu_B {\textstyle{\partial B \over \partial z}}\hat z\]

The assumption of intrinsic spin thus predicts that the atomic beam will split 
into 2 groups in the $z$ direction.

Classically (pre-quantum theory), the spin of the atom
may be at any angle to the 
$z$ axis.  The component of the magnetic moment along the $z$ axis is then {\bf 
any value} between $+{1\over2}\mu_0$ and $-{1\over2}\mu_0$ where $\mu_0$ is the
Bohr magneton.

The probability of the magnetic moment being between angles $\theta$ and 
$\theta + d\theta$ is equal to 
\begin{eqnarray*}
&=&{\hbox{area of section of sphere between $\theta$ and 
$(\theta+d\theta)$} \over \hbox{full area of sphere}}\\
&=& {(rd\theta)~(2\pi r\sin\theta) \over 4\pi r^2}\\
&=& {\sin\theta\, d\theta \over 2}\quad\raisebox{-5pt}{.}
\end{eqnarray*}
Hence the number of atoms with the $z$ component between $\mu$ and $\mu + d\mu$ 
is proportional to $\sin\theta\, d\theta$.


This fraction of the atoms will give magnetic moments between:
$${e \hbar \over 4m} \cos\theta~\hbox{and}~{e \hbar \over 4m} \cos (\theta + d
\theta)$$
{\em i.e.}, between:
$$\mu~{\rm and}~\mu -d\mu$$
where:
$$d\mu=-{d\over d\theta}\left({e\hbar\cos\theta\over 4m}\right)d\theta=
{e\hbar\over4m}\sin\theta\,d\theta.$$
Hence $d\mu$ is {\bf also} proportional to ($\sin\theta\,d\theta$).  The number 
of atoms between $\mu$ and $\mu-d\mu$ is thus proportional to $d
\mu$.  The classical prediction is therefore that the $z$ components of the 
magnetic moment will be uniformly distributed from $-{e \hbar \over 4m}~{\rm
to}~ +{e\hbar \over 4m}$.\par
\medskip 
\begin{flushleft}
\underline{\bf $z$ Displacement}
\end{flushleft}

An atom moving with velocity $v$ in the $x$ direction will be acted on by a 
force $-\mu_B \vec \nabla B$ for a time $t = {d_1 / v}$, where $d_1$ is the 
distance travelled by the atom in the magnetic field.  The acceleration along 
the $z$-axis (the direction of $\vec \nabla B$) will be 
$$\vec a_z = {\mu_B \vec \nabla B \over M}$$
where $M$ is the atom's mass; the velocity and deflection of the atom in this 
direction as it leaves the magnetic field will be:
\begin{eqnarray*}
\vec v_z &=& \vec a_z t \\
                    &=& {-\mu_B \vec \nabla B~d \over Mv}
\end{eqnarray*}
and
\[\vec s_z\,\raisebox{3pt}{$'$}=\vec a_zt^2/2={-\mu_B\vec\nabla Bd_1^2\over2Mv}.\]

From that point to the detector, $\vec v_z$ remains constant, so that the 
deflection, $\vec s$, at the detector is:
\begin{eqnarray*}
\vec s &=& \vec v_zd_z/v + \vec s_z\,\raisebox{3pt}{$'$} \\
&=& -\mu_B (\vec\nabla B) {[d^2_1 + 2d_1d_2] \over 2Mv^2}
\end{eqnarray*}
where $d_2$ is the distance between the magnet exit and the detector. For fixed 
$v,~ \vec s$ is proportional to $\mu_B$, and can be used to determine the 
distribution of $\mu_B$. Quantum mechanically, $\mu_B$ can be only $+\mu_0$ or 
$-\mu_0$, so $s$ will take on only two values, $+s_0$ and $-s_0$. Half of the 
atoms in the beam will arrive at each position.  Classically the atoms would be 
distributed uniformly among all values of $s$ between $+s_0$ and $-s_0$.

The Stern-Gerlach experiment, like many experiments utilizing molecular beam 
techniques, is limited to a certain extent by the fact that the atoms or 
molecules in a beam issuing from an oven at absolute temperature $T$ do not 
have a unique velocity.  The velocity distribution of the beam intensity is 
\[I(v)dv = 2I_0(v/\alpha)^3~~e^{-(v/\alpha)^{2}}~d(v/\alpha),\]
where $I_0$ is the total beam intensity, and $\alpha = \sqrt{2kT/M}$ is the 
most probable velocity of atoms in the oven (but not the beam). $k$ is the 
Boltzmann constant. 

Now if $s_\alpha$ is the deflection for an atom of velocity $\alpha$, then
\[s/s_\alpha = (\alpha/v)^2,\]
and changing variables in the equation above from $v/\alpha$ to $s/s_\alpha$ 
yields the following relation for the deflection pattern produced by a narrow 
beam of thermal atoms having moment $\mu_B$:
\[I(s)ds = aI_0 {s^2_\alpha \over s^3} e^{-(s_\alpha/s)} ds\]
where the half width of the undeflected beam is $a,\ a\ll s_\alpha$, and its 
intensity is $I_0$. For non-zero $a$, we must integrate the above equation over 
the width of the undeflected beam.  The resulting pattern, for $a = 
s_\alpha/10$, is shown in Fig.\ \ref{fig1}. 
A peak occurs at $s_{max}$, which equals 
$s_\alpha /3$.

Each value of $\mu_B$ gives rise to a pattern similar to Fig.\
\ref{fig1}, but with a 
different value for $s_\alpha$ and therefore for $s_{max}$.  The two values of 
$\mu_B$ given by quantum theory produce two patterns, one as shown, and one its 
mirror image. Classically an infinite number of values of $\mu_B$ occur, and 
the resultant composite pattern could be
a roughly bell-shaped curve centered at $s = 
0$.\par
\begin{figure}
\begin{center}
\epsfbox{stern_fig1.eps}
\caption{Deflection pattern for atoms having a unique value of $\mu_B$. 
Undeflected beam width, $a$, is $s_\alpha/10$.}\label{fig1}
\end{center}
\end{figure}
\medskip
\begin{flushleft}
\underline{\bf Summary of Predictions of $z$ direction distributions}
\end{flushleft}
\begin{enumerate}
\item
The classical (pre-quantum theory) predicts a uniform distribution 
between the limits. This is smeared by the thermal velocity distribution.
\item
The Schr\"odinger Quantum Mechanics (no spin) predicts an {\bf odd} 
number of groups. In the case of Potassium it predicts one group.
\item
The Dirac Theory predicts intrinsic spin.  In the case of Potassium 
it predicts {\bf two} groups.
\end{enumerate}
\par
\medskip
\begin{flushleft}
\underline{\bf The Apparatus}
\end{flushleft}
\medskip
\begin{flushleft}
\underline{\bf Potassium Oven}
\end{flushleft}

A slug of potassium has been sealed into the oven by a steel plug and 
sealed with a copper gasket.
The oven is heated by a nichrome heater and requires little 
power to keep it hot since it is thermally isolated on 4 mounts.

The oven is operated at 120$^\circ$ C (above the melting point at 
62.3$^\circ$C) 
and delivers a hot potassium vapor beam by means of a small hole 
covered by two stainless steel precision slits.

The heater has been carefully located with respect to the slits so that the 
slit area is kept hotter than the bulk storage chamber and is therefore
kept clean.\par
\medskip
\begin{flushleft}
\underline{\bf Oven Heater}
\end{flushleft}

The oven heater is driven from a stabilized 5 volt 5 Amp DC power supply.  The 
potassium atomic beam intensity depends upon the vapor pressure of the 
potassium and this depends critically upon the temperature.  The stabilized 
heater thus ensures a steady beam after a warm-up period.\par
\medskip
\begin{flushleft}
\underline{\bf Monitoring the Oven Temperature}
\end{flushleft}

The temperature of the oven is monitored by an iron-constantan thermocouple 
which passes through the oven mounting face.  
The thermocouple is connected to a voltmeter calibrated in $^\circ$C. The
approximate vapor pressure of potassium is $10^{-6}$ Torr at $63^\circ$C and
$10^{-3}$ Torr at $161^\circ$C.
\par\medskip
\begin{flushleft}
\underline{\bf Vacuum System}
\end{flushleft}

A pressure of less than $5\times10^{-6}$mm Hg (Torr) is 
necessary for a good signal since the 
potassium atoms can be easily scattered. The system is pumped by a fore-pump
and a 3 stage oil diffusion pump.

A water cooled cold trap is installed to reduce the unwanted migration of oil 
vapor.
The vacuum is monitored by two triode vacuum gauges-one on either side of the 
magnet box.  Only one gauge is needed for normal operation-the other has been 
used when locating leaks.

Migration of unwanted potassium through the vacuum system from the oven to the 
detector is reduced by baffles and by running the oven at low output. 
The baffles force 
any potassium atom travelling from the oven to the detector to strike at least 
one surface before reaching the detector unless the atom is travelling along 
the prescribed path of the potassium beam.

Tempered pyrex glass pipe forms the major part of the vacuum 
system and allows visual monitoring of the position of the beam gate, the 
temperature and position of the detector, and the possible accidental 
contamination of the oven end of the system by potassium or pump vapors. 
Connection between pipe and mating surfaces is made using a cast metal flange, 
a fiber ring to cushion the glass from the flange, and a neoprene O-ring which 
forms a vacuum seal between the pipe and the flat mating surface.
\par\medskip
\begin{flushleft}
\underline{\bf Beam Gate}
\end{flushleft}

The beam gate is attached to the long vertical baffle seen in Fig.\ 2 in the 
cross tube.  It is able to pivot and stay in position, either open or closed, 
since its center of gravity is higher than the pivot point.  The gate is moved 
from one position to another by a hand-held magnet outside the vacuum system, 
and rests against the glass in both positions, either fully open or fully 
closed.  When the gate is closed, the oven end of the system is virtually 
separated from the detector end of the system.

The overall layout of the potassium beam path is shown in Fig.\ 3.

\begin{figure}
\begin{center}
\epsfbox{stern_fig2.eps}
\caption{}%Figure 2
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfbox{stern_fig3.eps}
\caption{}%Figure 3
\end{center}
\end{figure}

\begin{flushleft}
\underline{\bf Vacuum System Maintenance}
\end{flushleft}

The maximum operating pressure of the Stern-Gerlach apparatus is approximately 
5.0 $\times 10^{-6}$mm of Hg. At higher pressures the potassium beam is
scattered
by the residual gases and no beam is detected.

The present system, when uncontaminated will maintain a vacuum around 
1.0 -- $2.0\times10^{-6}$mm of Hg with the oven off.

Due to the finite pumping speed through the pole piece box, the pressure in the 
detector end will be slightly higher than the pressure in the oven end.

\renewcommand{\labelenumi}{\alph{enumi}.}
\renewcommand{\theenumi}{\alph{enumi}.}
It is important to remember that:
\begin{enumerate}
\item
the ion gauges are never to be operated at pressures above 1.0 
$\times ~10^{-4}$mm of Hg.
\item
the grid current can be reduced by a factor of [10] to lengthen 
filament life; (in this case the pressure reads lower by factor of [10] also).
\item
the diffusion pump must have a constant cold water supply. 
\item
the diffusion pump must also always be on for normal operation.

The pressure will rise rapidly if the diffusion pump is shut off.
\item
the diffusion pump back pressure will always be greater than the 
pressure of the main system
\item
the mechanical forepump must operate {\bf continuously} to evacuate 
the gases pumped by the diffusion pump.

The rubber hose from the forepump to the diffusion pump should be free 
from crimps or sharp bends that would restrict the flow of gas inside.
\item
The instructor should be consulted for a vacuum system shutdown.
\end{enumerate}
\newpage
\begin{flushleft}
\underline{\bf Triode Ion Gauges}
\end{flushleft}

The 1949 
ion vacuum gauge
contains a plate, a grid, and two tungsten filaments, one of which is 
a spare. The filament is operated at a voltage such that all of the emitted 
electrons are drawn to the grid.  The grid is operated at a positive potential 
($V_p$). Electrons from the filament are accelerated to the positive grid. They 
bombard and ionize some of the molecules of any gas present in the tube.  The 
resulting positive ions are attracted to the negative plate and constitute the 
plate current ($I_p$). The ratio of this current to the grid current ($I_c$) is 
proportional to the gas pressure for pressures below about 0.0001mm Hg 
($10^{-4}$ Torr).  The pressure meter reads correctly for a 10 mA grid current.
\par
\medskip
\begin{flushleft}
\underline{\bf Nonuniform Magnetic Field}
\end{flushleft}

A cross section of the pole tips is shown in Fig.\ \ref{fig4}.
The iron pole tips
consist of a .218 inch diameter convex half cylinder spaced approximately 0.155
inch (3.97mm) from a .500 inch (1.27cm) diameter concave half cylindrical 
groove.

The concave circle, if extended, intersects the convex circle at the ends of 
its vertical diameter. These two curves are magnetic equipotentials of the 
magnetic field that would be present if the pole tips were replaced by two 
current carrying wires centered on the intersection points of the circles.  The 
magnet is designed to approximate this field, which has a reasonably 
homogeneous gradient in the region the beam is allowed to pass through.

\begin{figure}
\begin{center}
\epsfbox{stern_fig4.eps}
\caption{View of the pole unit looking ``down the beam path" with one end piece 
removed.  The extension of the concave circle ($r_1 = .25$ inches) intersects
the ends of the vertical diameter of the convex circle ($r_2 = .218$ inches). 
The dimensions of the spacings are $h_1 = .055$ inches and $h_2 = .100$
inches.}
\end{center}
\end{figure}

Field strength. A magnet capable of producing 0.4 Tesla (4000 gauss) across a 
.750 inch (1.91 cm) gap with a face 4 inches (10.16 cm) in diameter will 
produce the best results (this corresponds to about 7000 ampere turns), 
although respectable Stern-Gerlach patterns  can be obtained with half that 
strength. A 7000 ampere turn magnet produces a gradient $\vec\nabla B$ of a 
little over 100 Tesla/meter. 
\par\medskip
\begin{flushleft}
\underline{\bf Hot Wire Detector}
\end{flushleft}

The hot wire detector is an important advance over the techniques used in the 
original Stern-Gerlach experiment.  With it, the beam can be continuously 
sampled and experimental runs can be completed in a short time.  The detector 
is a hot pure tungsten filament wire, .005 inch in diameter, surrounded by a 
collecting cylinder kept at a voltage of about 15 volts below the wire.  
The collecting cylinder has
narrow slits to admit the potassium atoms.  
The wire is heated by a variable stabilized voltage supply which ``floats" at 
the bias voltage of 15 volts.

The detector can be moved laterally, by means of a micrometer, from outside the 
system; its position is indicated by the scale on the micrometer drive.  The 
micrometer shaft passes through an O-ring into the vacuum chamber.
\par\medskip
\begin{flushleft}
\underline{\bf Principle of Operation}
\end{flushleft}

The potassium atoms ionize when heated at the surface 
of wire.  A voltage placed across the collector and the hot wire will collect 
the positive ions if the collector is negative with respect to the hot wire; 
the small current is detected by a picoammeter or electrometer. 

The collector and the hot wire together are similar to a vacuum diode with a 
directly heated cathode.  In the detector, the ``plate" is always negative with 
respect to the ``cathode." In fact, if the bias voltage were reversed, the 
detector would act like a conducting diode; the collector would collect the 
electrons from the potassium and, in addition, it would collect electrons from 
the electron gas surrounding the hot wire.

The picoammeter current obtained is dependent on the filament temperature. A 
low filament current causes a low temperature and hence a low ionization rate 
and hence sluggish picoammeter readings. A high filament current will shorten
the lifetime of the filament.  DO NOT EXCEED 2.0 AMPS AT ANY 
TIME. A current of 1.5 amps is recommended.

The ionization rate increases temporarily whenever the filament temperature is 
increased slightly as absorbed potassium atoms are boiled off.  For this reason 
you should wait for about 5 minutes after changing the filament current before 
trusting the picoammeter readings.

The setting of the bias voltage is not critical. A voltage of 15 volts is 
recommended. 

A micrometer is provided to sweep the detector across the beam.  DO NOT EXCEED 
220 mils. The micrometer screw rod forms part of the seal in maintaining a good 
vacuum.  If the rod is pulled out farther than 220 mils the rod could slip out 
of the O-ring and this could destroy the vacuum and may damage the diffusion 
pump oil.
\par\medskip
\begin{flushleft}
\underline{\bf Operating Conditions}
\end{flushleft}

\renewcommand{\labelenumi}{\arabic{enumi}.}
\renewcommand{\theenumi}{\arabic{enumi}}
\begin{enumerate}
\item
Heat the oven at 20 W to an operating temperature of 120$^{\circ}$C. Back
off in heater current 
to $\sim$2.5 A when close to 120$^{\circ}$ so you don't overshoot.
Stabilize the temperature at 120$^{\circ}$.  It is
very important to make the actual measurements with a stable oven temperature.
\item
The recommended  hot wire current is $\sim$1.5 A.
The detector wire should be baked at  $\sim$2 A for 10 min. Expect a 
cool down of $\sim$5 min. before stable operation is achieved.  
\item
This experiment requires a good vacuum. Good signals will be obtained when the 
starting pressures before the oven bake are $\leq 2\times 10^{-6}$ Torr. 
\item
The signal current at the undeflected beam peak with a good vacuum 
can be as high as 10 pA. Depending on the conditions of the system it 
may be less. 
\end{enumerate}
\vspace{.25in}
\begin{flushleft}
\underline{\bf Procedure}
\end{flushleft}

\begin{enumerate}
\item
Set up according to the above conditions and scan for the signal with the 
magnet off. Once you know where to look you should do a first pass to locate 
the peak. 
\item
At the peak, optimize the detector hot wire current for maximum \underbar{
useful} signal. The voltage bias is not critical.
\item
Do a fine scan ($\sim$ 0.010{\tt "} steps) over the peak. Each point will probably
require both closed and open shutter readings. Check reproducibility of the 
data.
\item   
Do the same with the magnet on at $\sim$2.0 A.  
\item
Possible additional studies include a different oven temperature and/or a
different magnet setting.
\end{enumerate}
\vskip .25in
\begin{flushleft}
\underline{\bf Physics Analysis and Questions}
\end{flushleft}

\begin{enumerate}
\item
Calculate the magnetic field gradient from your data assuming the predicted
magnetic moment for Potassium. This will involve making some geometry 
measurements and deciding on a procedure for analyzing the data. 
\item
Derive an expression for the most probable velocity of the potassium atoms 
in the beam. Evaluate this quantity numerically at the measured 
operating temperature. Compare to the most probable value of the velocity
in the oven. Why are they different? 
\end{enumerate}
\end{document}



