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\begin{document}
D. W. Kerst Phys. Rev. {\bf 60,} 47 \hfill {\large \bf 1941}\\
\vspace{2cm}
\begin{center}
{\bf \Large The Acceleration of Electrons by Magnetic Induction}
\end{center}
\begin{center}
D. W. KERST\footnote{On leave at the General Electric Company Research
Laboratory.}\\
University of Illinois, Urbana, Illinois\\
(Received April 18, 1941)
\end{center}
\begin{abstract}
Apparatus with which electrons have been accelerated to an
energy of 2.3 Mev by means of the
electric field accompanying a changing magnetic field is described.
Stable circular orbits are formed
in a magnetic field, and the changing flux within the orbits
accelerates the electrons. As the magnetic
field reaches its peak value, saturation of the iron supplying flux through the orbit causes the
electrons to spiral inward toward a tungsten target. The $x$-rays
produced have an intensity
approximately equal to that of the gamma-rays from one gram of
radium; and, because of the
tendency of the $x$-rays to proceed in the direction of the
electrons, a pronounced beam is formed.
\end{abstract}
In the past the acceleration of electrons to very high voltage
has required the generation of the full voltage and the
application of that voltage to an accelerating tube
containing the electron beam. No convenient method for
repeated acceleration through a small potential difference
has been available for electrons, although the method has
been highly successful in the cyclotron for the heavier
positive ions at velocities much less than the velocity of
light.
Several investigators\footnote{G. Breit and M. A. Tuve, Carnegie
Institution Year Book (1927-28) No. {\bf 27,} p. 209.}
\footnote{R. Wider\"oe, Arch. f. Electrotechnik {\bf 21,} 400 (1928).}
\footnote{E. T. S. Walton, Proc. Camb. Phil. Soc. {\bf 25,} 469-81
(1929).} \footnote{W. W. Jassinsky, Arch. f. Electrotechnik {\bf 30,}
500 (1936).} have considered the possibility
of using the electric field associated with a time-varying
magnetic field as an accelerating force. This is a very
attractive possibility because the magnetic field can be
used to cause a circular or spiral orbit for the electron while
the magnetic flux within the orbit increases and causes a
tangential electric field along the orbit. The energy gained
by the electron in one revolution is about equal to the
instantaneous voltage induced in one turn of a wire placed
at the position of the orbit. Since the electron can make
many revolutions in a short time, it can gain much energy.
The comparatively small momentum of a high energy
electron requires correspondingly small values of $Hr$ for
high energy orbits. For example, the energy of an electron
when $v \sim c$ is $KE = 3 \times 10^{-4} Hr - 0.51$ million
electron volt. Thus with $H=3000$ oersteds
and $r = 5$ cm, the energy of the electron would be about 4
Mev, and the orbit could be held between the poles of a
small magnet.
Because of the experimental experiences of previous
investigators$^{1-3}$ with this method of acceleration, a rather
detailed study of the focusing to be expected was made,
and it is presented m the paper immediately following this
one. With the results of this theoretical investigation to
guide the design, it was possible to make an induction
accelerator which produced $x$-rays of 2.3 Mev\footnote{D. W. Kerst,
Phys. Rev. {\bf 58,} 841 (1940).} \footnote{D. W. Kerst, Phys.
Rev. {\bf 59,} 110 (1941).}. Briefly, in the focusing theory it is shown that:
1. The electrons have a stable orbit, ``equilibrium orbit,''
where
\begin{equation}
\phi_0 = 2 \pi r_0^2 H_0.
\end{equation}
$\phi_0$ is the flux within the orbit at $r_0$, and $H_0$ is the
magnetic field at $r_0$. Both $\phi_0$ and $H_0$ are increased during
the acceleration process. This flux condition holds for all
velocities, of the electrons, and it shows that if a maximum
flux density of 10,000 gauss is allowed in the iron then
5000 oersteds is the maximum magnetic field which can be
used at the orbit.
2. In the plane of their orbits the electrons oscillate
about their instantaneous circles, circles for which
$p = eHr/c$, with an increasing frequency
\begin{equation}
\omega_r = \Omega(1 - n)^{1/2},
\end{equation}
where $\Omega$ is the angular velocity of the electron in its
orbit, and $\omega_r$ is $2 \pi$ times the radial focusing
frequency. The number $n$ is determined by the
radial dependence of the magnetic field, which we take to
be of the form $H \sim 1/r^n$. For radial focusing $n$ must be less
than unity.
3. Axial oscillations, oscillations perpendicular to the
plane of the orbit, have
\begin{equation}
\omega_A = \Omega n^{1/2}.
\end{equation}
For axial stability $n$ must be greater than zero. If the beam
is to be smaller in an axial direction than it is in a radial
direction then $n > 1/2$.
4. Decrease of the amplitude of both axial and radial
vibration occurs because of the increase of the restoring
force with increasing magnetic field.
At nonrelativistic velocities the damping is
\begin{equation}
da/a = - dE/4E,
\end{equation}
where $dE/E$ is the fractional increase of the kinetic energy
of an electron and $da/a$ is the fractional decrease in
amplitude of the oscillation about the instantaneous circle.
This holds for both axial and radial oscillations.
5. Instantaneous circles not coincident with the
equilibrium orbit shrink or expand toward coincidence:
\begin{equation}
dx/x = - dE/2E,
\end{equation}
where $x$ is the displacement of the instantaneous circle from
the equilibrium orbit and $dx$ is the shift of the circle toward
the equilibrium orbit while the electron's energy increases
by the fraction $dE/E.$
Because of the shrinking or expansion of the
instantaneous circle of a displaced electron toward the
equilibrium orbit and the decrease of the amplitude of
oscillation of an electron about its instantaneous circle, it
was expected that a cathode or an electron injector placed
outside of the equilibrium orbit could shoot in electrons
which would miss the injector on successive revolutions
around the magnet. Furthermore, since an instantaneous
circle for electrons with a constant injection energy exists
within the acceleration chamber for a small but finite
interval of time while the magnetic field is increasing, a
finite amount of charge should be captured in orbits not
striking the walls. With constant potential on an injector the
electrons first would hit the outer wall of the chamber
before the magnetic field had grown large enough to give
the electrons a radius of curvature less than the
radius of the wall. Then as the field increased it would
reach values which give instantaneous circles within the
vacuum tube. Eventually the field would be so large that
the electrons coming out of the injector strike the walls or
spiral around in small circles.
From Eq. (2) it can be seen that the spreading rays from
the injector will form injector images $\pi(1-n)^{-1/2}$ radian
apart, since if each ray of the beam oscillates about the
instantaneous circle, it also oscillates about the central ray
from the injector.
Equations (4) and (5) indicate that for large da or $dx$ it is
desirable to use a small injection voltage $E,$ and a high
voltage per turn $dE.$ A shift $da$ in two revolutions or $dx$ in
one revolution of about 1 mm when the displacement of the
injector from the equilibrium orbit is $x = a =1.5$ cm would
require $dE/E \simeq \frac{1}{8}$. It is not possible to decrease
the injection voltage $E$ indefinitely since scattering increases when this is done, and it may be that the beam would be lost by scattering
through an angle greater than that which magnetic
focusing can handle. Conservative estimates on the
scattering out of a cone of $7_{\circ}$ half-angle, which is about the
focusing limit, showed that at $10^{-6}$ mm of Hg air pressure in
the tube, a maximum $Hr = 15,000$ gauss cm, and $f=600 $
cycles/sec, for the frequency of oscillation of the magnetic
field, the injection voltage must exceed 100 volts for less
than a 20-percent loss of beam. This then requires that $dE$
or the voltage per turn at the equilibrium orbit must be at
least 12 volts at the time of injection. In the magnet which
was built this condition was easily satisfied, for 25 volts per
revolution could be attained. There is no trouble with
energy loss by the electrons in their long paths through the
residual gas in the vacuum tube. Wider\"oe$^2$ was aware of the
flux requirement (1), but apparently he did not realize that
the voltage gain per revolution could not be too small
compared with the injection voltage, or that it would be
necessary to inject the electrons nearly tangent to the
equilibrium orbit.
After the injected electrons have orbits approximately
coincident with the equilibrium orbit, they should remain
close to $r_0$. A simple method was used to shift the
equilibrium orbit so that the electrons struck a thin tungsten
target.
A portion of the flux through the center of the orbit passed
through disks cemented to the center of the pole faces and
made of material which saturated easily. As saturation
progressed, the electrons had to move inward to a smaller
radius of curvature in order that $\phi = 2 \pi r^2 H$. Eventually the
electrons grazed past the target producing thin
target $x$-radiation.
\section*{
Space Charge Effect}
~~~It is difficult to estimate the effect of space charge
within the beam on the formation of images, since there are
two focusing oscillations, axial and radial, in general
having different frequencies. However, an estimate can be
made of the upper limit of current which the magnetic
forces can hold.
At injection the magnetic forces are small so that the
estimate should be made with the magnetic field at this
time. The electrons at the edge of the beam, which is
assumed to be a cylinder of radius $\Delta$ determined by the
distance between the equilibrium orbit and the injector,
will experience a repulsive force of
\begin{equation}
F_e = (c Q e)/(10 \pi r_0 \Delta)
\end{equation}
dynes. $Q$ is the number of coulombs in the orbit, and $r_0$ is
the radius of the orbit.
For radial unbalanced magnetic forces we get
\begin{equation}
F_m = e^2 H^2(1-n) \Delta / mc^2
\end{equation}
dynes acting on the electrons toward $r_0$. If $F_e = F_m$ at the
edge of the beam, then this balance will hold within the
beam for all other distances $\delta$ smaller than $\Delta$ because the
Q of (6) is proportional to $\delta^2$. Thus
\begin{equation}
I=10 \pi f e r_0 \Delta^2 H^2 (1-n)/mc^3.
\end{equation}
Or in terms of the injection voltage $E,$
\begin{equation}
I= \pi \Delta^2 fE (1-n)/(15 r_0 c).
\end{equation}
$I$ is the target current in amperes, $f$ the frequency of
oscillation of the magnetic field, and $n$ gives the
dependence of the magnetic field on $r$. Since the value of $E$
which can be used depends upon the voltage per turn $dE,$
which is also dependent on the peak magnetic field
reached or the output energy, and also upon the orbit shift
$da$ which is required to miss the injector, we get
\begin{equation}
I= 2 \pi^3 \Delta^3 f^2 (KE + \frac{1}{2})(1-n)/(45 cda \times 10^4),
\end{equation}
where $KE$ is the final energy of the electrons in Mev.
Inserting into (10) the constants of the machine which was
built, we have $\Delta = 1.5$ cm, $f=600$ cycles/sec., $KE=2.3$ Mev,
and $da \simeq 0.1$ cm. The target current should be about 0.03
microampere. This is fairly close to the lower limit of
current as estimated from the $x$-ray output of the
accelerator. This lower limit was found by using the thick
target yield of Van Atta and Northrup\footnote{L. C. Van Atta and
D. L. Northrup, Am. J. Roentgen.
Rad. Ther. {\bf 41,} 633 (1939).} in the direction of
the electron beam for comparison with the $x$-ray yield from
the induction accelerator. It is found that the target current
in the induction accelerator must be greater than 0.02
microampere. It is only possible to find the lower limit of
this current since the output is thin target radiation.
A curious behavior which was immediately noticed
when the apparatus began to work was that injection
voltages much greater than the largest voltage which
would allow the orbit to miss the injector could be used. In
fact the yield increased greatly as this voltage was raised.
This seems understandable on the basis of space charge
forces spreading the beam away from the central ray of the
injector.
At relativistic energies space charge forces are
completely balanced by magnetic self-focusing of the
beam, for the electric force on a stray electron at a distance
$\Delta$ from the beam center is
\begin{equation}
e {\cal E} = 2 \sigma e/\Delta,
\end{equation}
where $\sigma$ is the linear charge density in e.s.u./cm. The
magnetic attraction due to the main current in the beam is
\begin{equation}
evH/c = (v/c)^2 2 \sigma e /\Delta.
\end{equation}
Thus it is evident that when $v \rightarrow c$, the magnetic
pull of the beam for a stray electron just equals the electrostatic
repulsion. Or, from the point of view of an observer on the
electron, the spacing of the fixed number of electrons
around the orbit will increase, since as $v \rightarrow c$ his yardstick becomes a smaller fraction of the circumference of the orbit.
A photograph of the apparatus is shown in Fig. 1.
For the production of the rapidly changing magnetic
field, with the proper dependence on the radius, finely
laminated iron pole faces were made from 0.003-inch
silicon steel sheets. The return magnetic circuit made from
the same material was interleaved for mechanical strength;
and the roughly circular centra) pole pieces were
\begin{figure}
\centerline{\resizebox{10cm}{!}{\includegraphics{fig1.gif}}}
\caption{ The induction accelerator. The glass doughnut is
between pole faces which are held apart by eccentric
wedges.}
\end{figure}
formed by stacking with different widths of laminations as
shown in Fig. 2. Each pole piece was capped by a disk of
radially arranged laminations so that perfect circular
symmetry was achieved at the pole surfaces ($B,$ Fig. 2). The
whole pole face was held together by a thick Transite or
asbestos board ring about its perimeter, with cement of
water glass and flint dust filling the cracks between the
laminations and hardened by baking. The pole caps were
held against the pole pieces by eccentric wedges between
the Transite rings.
To supply the flux within the orbit which was necessary
to hold the electrons out at the equilibrium orbit, two disks
about two inches in diameter, each made of pressed iron
dust of permeability about eight, were cemented onto the
flat center portions of the pole faces. The thickness of these
disks was chosen so that the equilibrium orbit was formed
at about 7.5 centimeters radius, and the final adjustment of
the position of the equilibrium orbit was made by painting a
mixture of water glass and iron filings, or the turnings from
the compressed iron dust disks, onto the surface of the iron
that the reluctance of the gap was correct. Since the iron
particles in these disks are separated, the flux density
through them is greater than the average flux density
through the disks. This causes the saturation which shrinks
the equilibrium orbit down to the target.
Because of the large leakage flux the main coil of the
magnet had to be highly subdivided. Two hundred strands
of No. 20 double enameled wire were made into a cable
approximately 10 yards long which was twisted about ten
times and which had its inside wires interchanged with its
outside wires several times. This was formed into a 10-turn
coil with about $\frac{1}{32}$ inch of insulation between turns. Two
such 10-turn coils were made and fitted with large lugs of
1-inch copper tubing. The coils were wrapped with cotton
tape, dipped in Bakelite varnish and baked. Figure 3 shows
the circuit with the coils connected to a total of eighty
5-microfarad Pyranol condensers which were rated at 660
volts a.c. Energy was supplied to this resonating circuit by
a 2-turn primary of stranded enameled wire around the pole
pieces. The r.m.s. electromotive force in this primary
\begin{figure}[h]
\centerline{\resizebox{10cm}{!}{\includegraphics{fig2.gif}}}
\caption{
Dimensions of the magnet. Parts $B$ are the pole
caps made with all laminations placed in a radial direction.
The iron dust disks supply the central flux. Outside of the
14$^{\circ}$ conical surface a flat rim on the pole face tends to
prevent too rapid decrease of the field at the edge of the
gap.}
\end{figure}
could be run up above 100 volts, but 60 to 80 volts were
usually used in operation since this was sufficient to
saturate the powdered iron and to collapse the orbit. Power
was supplied by a 4-kilowatt 600-cycle alternator driven by
a direct-current motor with an adjustable speed.
To determine how the magnetic field decreased with $r$ a
small search coil was arranged so that it could be held
between the poles at different radial distances from the
center. The e.m.f. from
this search coil was bucked against the e.m.f. from a
voltage divider connected across a 1-turn coil about the leg
of the magnet, equality of e.m.f. being determined by no
deflection of the beam of an oscillograph. Since the wave
form of the voltage on the search coil differed slightly from
the wave form from the coil around the core, balance could
be obtained only at one phase, and the phase of interest was
that of zero magnetic field. The use of the oscillograph as a
null instrument in this way proved to be sufficiently
accurate. The pole faces gave a field following the $1/r^{2/3}$ law
between a radius of 4.5 centimeters and 9.25 centimeters
when the separation between the flat central portions of the
pole faces was 2.8 centimeters. At $r = 10$ cm the law was
$1/r.$
To determine the position of the equilibrium orbit use
was made of a concentric system of seven 1-turn coils set
in grooves in a Bakelite disk which fit snugly around one of
the iron dust disks. Relative values for the e.m.f.'s induced
in these coils were determined by the same null method
which was used for determining the shape of the field.
These data could be used in several ways to find $r_0$. The
simplest method is to find the radius of minimum electric
field, since this will be $r_0$.
\section*{
The Acceleration Chamber}
~~~It was desirable to have the volume available for
electron orbits as large as possible. This required a
doughnut-shaped glass vessel with walls parallel to the
conical pole faces. The outer diameter was 20 cm, and the
center of the doughnut was 7 cm in diameter. The glass
work for this vessel required great skill\footnote{I am indebted to
the Vacuum Tube Department of the
General Electric $X$-Ray Corporation for producing a
Correctly shaped tube.}. A 20-cm spherical
bulb with a wall thickness of 2.5 mm was flattened and
dished conically on both sides so that it would fit between
the magnet poles. The central 7-cm hole of the doughnut
was formed by pushing the centers of the dished faces
together and picking out the glass.
The inside of this bulb was coated with a thin
conducting layer by chemical silvering. This coating is
necessary to prevent stray charges from building up
potentials on the walls. Contact is made to the silver
coating which is then
\begin{figure}[h]
\centerline{\resizebox{12cm}{!}{\includegraphics{fig3.gif}}}
\caption{The resonant circuit which energizes the magnet.
Losses are supplied by a 4-kilowatt 600-cycle-per-second
generator.}
\end{figure}
grounded outside of the tube. Resistances of the silver coat
between 20 ohms and 300 ohms between test probes put in
the side arms were used.
Both target and injector were mounted on the glass
pinch seal which was waxed to a flare on the doughnut. The
target was a piece of \mbox{0.015-inch} tungsten sheet folded to
present an edge of large radius of curvature to the electron
orbit which shrinks inward toward it. Thin tungsten is used
so that eddy currents will not be excessive. The injector
was made of thin molybdenum sheets and a cylindrically
spiraled tungsten wire for the filament. The whole
assembly is shown in Fig. 4.
Fortunately the presence of this much metal near the
orbit does not seem to disturb the local magnetic field too
much. However, the orbits are destroyed if a piece of metal
is brought up to the side of the glass doughnut while the
machine is running. This was observed when a block of
beryllium was put near the tube for experiments with
photo-disintegration. The $x$-ray yield disappeared when the
beryllium was too close. The original injector was arranged
to send a beam of electrons in both directions around the
magnet so that both directions of field could be used. Thus
two beams of $x$-rays oppositely directed were produced.
\section*{Operation}
~~~~Although the focusing theory shows that for the present
design of the accelerator electrons should be injected at
approximately 200 volts, it was found that the yield
increased with voltage and that it was still increasing at 600
volts. Likewise the negative voltage on the small focusing
plates $G$ beside the filament gave an
\begin{figure}[h]
\centerline{\resizebox{14cm}{!}{\includegraphics{fig4.gif}}}
\caption{Cross section of the doughnut-shaped acceleration chamber. The
equilibrium orbit is at $r_0$. $T$ is the tungsten target, $A$ is the
injector, $B$ is a top view of the injector. A ribbon
beam of electrons from the filament $F$ is shot out through slots in the positive plates $P.$ $G$ are negative focusing electrodes.}
\end{figure}
increasing yield as it was raised to 200 volts. The beam
from the injector was allowed to pour into the vacuum tube
continuously.
Initially the yield obtained was approximately
equivalent to 10 millicuries of radium when measured in a
direction from the target at right angles to the electron
beam as it strikes the target, and the yield was
approximately ten times greater in the direction of the
electron beam. Later the yield in the direction of the beam
was increased to the equivalent of one gram of radium.
Figure 5 shows an absorption curve of the radiation as it
passes through lead, and the comparison with
two-million-volt $x$-rays from the M. I. T.
\begin{figure}[h]
\centerline{\resizebox{12cm}{!}{\includegraphics{fig5.gif}}}
\caption{ The relative intensity of $x$-rays from.the induction
accelerator as a function of absorber thickness. A is
absorption in copper. The absorption coefficient is 0.454
cm$^{-1}$ which gives a monochromatic equivalent of 1.35
Mev for the $x$-rays. $B$ is the curve for lead. The coefficient
is 0.62 cm$^{-1}$, which corresponds to 1.4 Mev
monochromatic equivalent. $C$ is the lead absorption curve
of Van Atta and Northrup for $x$-rays produced by 2.0-Mev
electrons striking a thick target.}
\end{figure}
electrostatic machine$^7$
shows that the energy reached in the induction accelerator is
approximately 2.0 million electron volts and that the
radiation produced is thin target radiation. With other iron
dust disks cemented on the pole faces, absorption
coefficients indicating 2.3-Mev $x$-rays have been obtained.
By means of a collimated Geiger-M\"uller counter the
source of the $x$-rays was shown to be the target. A large
lead block with a small hole in it let an abundant amount of
x-rays through to the Geiger-M\"uller tube only when it was
pointed at the target and not when it was pointed at the
injector or other portions of the vacuum tube. By
connecting the vertical deflection plates of an oscillograph
to the Geiger-M\"uller counter and the horizontal deflection
plates to a 1-turn coil around one leg of the magnet, the
phase of the magnetic field at which electrons struck the
target could be determined. As was expected it was found
that the greater the excitation of the dynamo which
energizes the magnet, the earlier in the cycle the iron dust
saturated and brought the beam in to the target. Lowering
the primary voltage postponed sufficient saturation in the
iron dust until the peak of the magnetic field was reached.
The Geiger-M\"uller counter then gave $x$-ray pulses at the
center of the oscillograph screen. This indicated a path
length of about sixty miles from injector to target. If the
primary voltage was lowered beyond this point, the yield
disappeared, for the
electrons were not drawn in to the target but were slowed
down by the decreasing magnetic field. Fortunately the
operation of the accelerator is not sensitive to the
alignment of the pole faces. No difference in the output can
be detected when the pole faces are placed off axis as far as
a thirty-second of an inch. It is also surprising that vacuum
requirements are not as severe as was expected. No
rigorous outgassing is necessary and the apparatus has
been run with a vacuum as poor as $10^{-5}$ mm Hg. The tube
can be opened for changes and operated three-quarters of
an hour after sealing shut.
At present, low flux densities have been used at the
orbit. When these are increased, it should be possible to go
to 5 million volts even with this small model. One of the
promising possibilities for the induction accelerator as a
research tool is that the electrons from the beam can come
out through the glass walls of the doughnut after they strike the
target. They should be fairly homogeneous in energy
provided that the target has a high atomic number. The
great increase in bremsstrahlung production with rising
electron energy in addition to the concentration of this
radiation in a cone of solid angle $mc^2/E$, about the original
electron direction gives the induction accelerator the
possibility of providing an intense source of $x$-radiation for
nuclear investigations. Since there is no evident limit on
the energy which can be reached by induction acceleration,
it may soon be possible to produce some small scale
cosmic-ray phenomena in the laboratory.
I am indebted to Professor H. M. Mott-Smith and
Professor R. Serber for many discussions of the theoretical
aspects of this problem and to Mr. R. P. Jones for
assistance in the construction of the magnet.
\end{document}
%ENCODED MAY 27, 2003 BY NIS;