A modern Cyclotron for radiation
therapy
A
cyclotron is a type of
particle accelerator. Cyclotrons
accelerate
charged particles using
a
high-frequency, alternating
voltage (
potential difference). A perpendicular
magnetic field causes the particles
to
spiral almost in a circle so that they
re-encounter the accelerating voltage many times.
Ernest Lawrence, of the University of
California, Berkeley, is credited with the development of the cyclotron
in 1929, though others had been working along similar lines at the
time. .
The
largest Cyclotron in the world is housed at the Tri-University
Meson Facility (TRIUMF) at the
University of
British Columbia, Vancouver, Canada, and is run as a consortium of
eleven Canadian universities and the National Research Council
Canada. The 18m diameter, 4000 tonne main magnet produces a
field of 0.46 T while a 23 MHz 94 kV electric field is used to
accelerate the 200 μA beam.
How the cyclotron works
Diagram of cyclotron operation from Lawrence's 1934 patent.
Beam of electrons moving in a
circle.
Lighting is caused by excitation of gas atoms in a bulb.
The
electrodes shown at the right would be
in the
vacuum chamber, which is flat,
in a narrow gap between the two
poles
of a large magnet.
In the cyclotron, a high-frequency
alternating voltage applied across the
"D" electrodes (also called "dees") alternately attracts and repels
charged particles. The particles,
injected near the center of the magnetic field,
accelerate only when passing through the gap
between the electrodes. The perpendicular
magnetic field (passing vertically through
the "D" electrodes), combined with the increasing energy of the
particles forces the particles to travel in a spiral path.
With no change in energy the charged particles in a magnetic field
will follow a circular path. In the cyclotron, energy is applied to
the particles as they cross the gap between the dees and so they
are accelerated (at the typical sub-relativistic speeds used) and
will increase in mass as they approach the speed of light. Either
of these effects (increased velocity or increased mass) will
increase the radius of the circle and so the path will be a
spiral.
(The particles move in a spiral, because a
current of electrons or ions, flowing
perpendicular to a magnetic field,
experiences a perpendicular force. The charged particles move freely in
a vacuum, so the particles follow a spiral path.)
The radius will increase until the particles hit a target at the
perimeter of the vacuum chamber. Various materials may be used for
the target, and the collisions will create secondary particles
which may be guided outside of the cyclotron and into instruments
for analysis. The results will enable the calculation of various
properties, such as the mean spacing between atoms and the creation
of various collision products. Subsequent chemical and particle
analysis of the target material may give insight into nuclear
transmutation of the elements used in the target.
Uses of the cyclotron
For several decades, cyclotrons were the best source of high-energy
beams for
nuclear physics
experiments; several cyclotrons are still in use for this type of
research.
Cyclotrons can be used to treat
cancer. Ion
beams from cyclotrons can be used, as in
proton therapy, to penetrate the body and
kill tumors by
radiation damage,
while minimizing damage to healthy tissue along their path.
Cyclotron beams can be used to bombard other atoms to produce
short-lived
positron-emitting isotopes
suitable for
PET imaging.
Problems solved by the cyclotron
The cyclotron was an improvement over the
linear accelerators that were available
when it was invented. A linear accelerator (also called a linac)
accelerates particles in a straight line through an evacuated tube
(or series of such tubes placed end to end). A set of electrodes
shaped like flat donuts are arranged inside the length of the
tube(s). These are driven by high-power radio waves that
continuously switch between positive and negative voltage, causing
particles traveling along the center of the tube to accelerate. In
the 1920s, it was not possible to get high frequency radio waves at
high power, so either the accelerating electrodes had to be far
apart to accommodate the low frequency or more stages were required
to compensate for the low power at each stage. Either way,
higher-energy particles required longer accelerators than
scientists could afford.
Modern linacs use high power
Klystrons and
other devices able to impart much more power at higher frequencies.
But before these devices existed, cyclotrons were cheaper than
linacs.
Cyclotrons accelerate particles in a spiral path. Therefore, a
compact accelerator can contain much more distance than a linear
accelerator, with more opportunities to accelerate the
particles.
Advantages of the cyclotron
- Cyclotrons have a single electrical driver, which saves both
money and power, since more expense may be allocated to increasing
efficiency.
- Cyclotrons produce a continuous stream of particles at the
target, so the average power is relatively high.
- The compactness of the device reduces other costs, such as its
foundations, radiation shielding, and the enclosing building.
Limitations of the cyclotron
The magnet portion of a large
cyclotron.
The gray object is the upper pole piece, routing the magnetic
field in two loops through a similar part below.
The white canisters held conductive coils to generate the
magnetic field.
The D electrodes are contained in a vacuum chamber that was
inserted in the central field gap.
The
spiral path of the cyclotron beam can
only "sync up" with klystron-type (constant frequency) voltage
sources if the accelerated particles are approximately obeying
Newton's Laws of Motion. If
the particles become fast enough that
relativistic effects become important,
the beam gets out of phase with the oscillating electric field, and
cannot receive any additional acceleration. The cyclotron is
therefore only capable of accelerating particles up to a few
percent of the speed of light. To accommodate increased mass the
magnetic field may be modified by appropriately shaping the pole
pieces as in the
isochronous
cyclotrons, operating in a pulsed mode and changing the
frequency applied to the dees as in the
synchrocyclotrons, either of which is
limited by the diminishing cost effectiveness of making larger
machines. Cost limitations have been overcome by employing the more
complex
synchrotron or
linear accelerator, both of which have
the advantage of scalability, offering more power within an
improved cost structure as the machines are made larger.
Mathematics of the cyclotron
Non-relativistic
The
centripetal force is provided
by the transverse magnetic field
B, and the force on a
particle travelling in a magnetic field (which causes it to be
angularly displaced, i.e spiral) is equal to
Bqv.
So,
- \frac{mv^2}{r} = Bqv
(Where m is the mass of the particle, q is its charge, B the
magnetic field strength, v is its velocity and r is the radius of
its path.)
The speed at which the particles enter the cyclotron due to a
potential difference, V.
- v = \sqrt{\frac{2Vq}{m}}
Therefore,
- \frac{v}{r} = \frac{Bq}{m}
v/r is equal to angular velocity,
ω, so
- \omega = \frac{Bq}{m}
And since the angular frequency is
- \omega = {2\pi} f
Therefore,
- f = \frac{Bq}{2\pi m}
But this is for one complete loop and cyclotron must switch twice
every cycle, therefore
- f_c = \frac{Bq}{\pi m}
This shows that for a particle of constant mass, the frequency does
not depend upon the radius of the particle's orbit. As the beam
spirals out, its frequency does not decrease, and it must continue
to accelerate, as it is travelling more distance in the same time.
As particles approach the speed of light, they acquire additional
mass, requiring modifications to the frequency, or the magnetic
field during the acceleration. This is accomplished in the
synchrocyclotron.
Relativistic
The radius of curvature for a particle moving relativistically in a
static magnetic field is
- r = \frac{\gamma m v}{q B}
- where
- \gamma=\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}} the
Lorentz factor
Note that in high-energy experiments energy, E, and momentum, p,
are used rather than velocity, and both measured in units of
energy. In that case one should use the substitution,
- \frac{E}{p} = v
- where this is in Natural
units
The relativistic cyclotron frequency is
- f=f_c\sqrt{1-\left(\frac{v}{c}\right)^2},
- where
- f_c is the classical frequency, given above, of a charged
particle with velocity
- v circling in a magnetic field.
The rest mass of an electron is 511 keV/c
^{2}, so the
frequency correction is 1% for a magnetic vacuum tube with a 5.11
keV/c
^{2} direct current accelerating voltage. The proton
mass is nearly two thousand times the electron mass, so the 1%
correction energy is about 9 MeV, which is sufficient to induce
nuclear reactions.
An alternative to the synchrocyclotron is the
isochronous cyclotron, which has a
magnetic field that increases with radius, rather than with time.
The de-focusing effect of this radial field gradient is compensated
by ridges on the magnet faces which vary the field azimuthally as
well. This allows particles to be accelerated continuously, on
every period of the radio frequency, rather than in bursts as in
most other accelerator types. This principle that alternating field
gradients have a net focusing effect is called
strong focusing. It was obscurely known
theoretically long before it was put into practice.
Related technologies
- The spiraling of electrons in a cylindrical vacuum chamber
within a transverse magnetic field is also employed in the magnetron, a device for producing high frequency
radio waves (microwaves).
- The Synchrotron moves the particles
through a path of constant radius, allowing it to be made as a pipe
and so of much larger radius than is practical with the cyclotron
and synchrocyclotron. The larger radius allows the use of numerous
magnets, each of which imparts angular momentum and so allows
particles of higher velocity (mass) to be kept within the bounds of
the evacuated pipe. The magnetic field strength of each of the
bending magnets is increased as the particles gain energy in order
to keep the bending angle constant.
See also
External links