Math Is Fun Forum

Q: What do you get if you divide the circumference of a bowl of ice cream by its diameter?
A: Pi a'la mode.
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Q: Why did the ice cream truck break down?
A: Because of the Rocky Road.
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A: After they have a very frank relationship!
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Q: What is a man's idea of a balanced diet?
A: A bag of potato chips in each hand!
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Q: How do you learn how to make ice cream?
A: In Sunday (Sundae) School.
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Cyclotron

Gist

A cyclotron is a compact particle accelerator that uses a constant magnetic field to bend charged particles into a spiral path and a rapidly varying electric field to accelerate them to high speeds. Invented by Ernest Lawrence in 1929–1930, it is primarily used in medicine to produce short-lived radioisotopes for cancer diagnosis (PET scans) and treatment. (PET : Positron Emission Tomography).

Cyclotrons are particle accelerators used in medicine (producing radioisotopes for imaging/therapy, cancer treatment via proton beams) and nuclear physics research (bombarding nuclei for experiments, studying atomic properties, creating new elements/isotopes). Their compact size makes them practical for generating high-energy particle beams for these scientific and medical applications, offering advantages over linear accelerators in certain scenarios.

Summary

A cyclotron is a particle accelerator that uses magnetic and electric fields to speed up charged particles to very high speeds and powers many of the tools, treatments, and discoveries that improve our daily lives.

If you have ever had a PET scan at a hospital or heard about radiation treatment for cancer or brain tumours, there's a good chance a cyclotron was involved.

But what is a cyclotron and how is it used?

Let’s break it down.

A cyclotron is a type of particle accelerator. It uses magnetic and electric fields to speed up charged particles like protons or ions to very high speeds. This allows the particles to collide with target materials to produce radioisotopes through nuclear reactions.

Radioisotopes have several uses, including in life saving medical treatments, scientific research, and even clean energy technologies.

The cyclotron was invented in 1931 by American physicist Ernest O. Lawrence and his student M. Stanley Livingston at the University of California, Berkeley. Their early prototype - just about 10 cm wide - was already capable of accelerating particles to high energy levels. Lawrence’s groundbreaking work earned him the Nobel Prize in Physics in 1939.

How does a Cyclotron Work?

The process begins when charged particles like positive or negative ions are injected into the centre of the cyclotron, where they start to move outward in a spiral path.

Inside the cyclotron, are two hollow, D-shaped metal electrodes called ‘dees’, placed between the poles of a large magnet. The magnetic field forces the particles into a circular path, while an alternating electric field boosts the particle’s energy every time it crosses the gap between two dees. As the particles gain speed and energy, they continue to spiral outward.

Once the particles reach the outer edge of the cyclotron, they are directed toward a target. When the accelerated particles collide with the target, they can cause nuclear reactions, producing radioactive isotopes.

Nearly a century after their invention, cyclotrons remain in high demand because of their reliability, efficiency, and versatility.

What’s the Difference Between Cyclotrons and other Particle Accelerators?

Particle accelerators have many applications in medicine, industry and research. These machines accelerate charged particles, such as electrons and protons, to high speeds, sometimes even close to the speed of light.

While all particle accelerators share a common goal - boosting the energy of particles - they achieve this in different ways.

Cyclotrons accelerate particles in a spiral path using a constant magnetic field and an alternating electric field. The spiral design is one of the cyclotron’s main advantages. It allows for continuous acceleration in a relatively small space. As a result, cyclotrons are typically smaller, often room-sized, and more affordable than other accelerators. They can be installed in hospitals or university labs without needing massive facilities. Cyclotrons are also well-suited for producing specific types of radioactive isotopes needed in medical imaging and cancer treatment, and for other localized applications in research or industry.

In contrast, linear accelerators, or linacs, propel particles in a straight line using a series of electric fields. While linacs can be simpler in design, they often require much more space to achieve the same energy levels as a cyclotron. They are commonly used in radiotherapy, where precise targeted beams of radiation are used to treat tumours.

Another type of accelerator is the synchrotron - a much larger and more complex machine found in national research centres. Like cyclotrons, they guide particles in a circular path, but with variable magnetic fields and radiofrequency acceleration. These machines can reach extremely high energies, making them suitable for research in particle physics, materials science, and even drug development. However, due to their size and cost, they are typically used by national or international research centres, not hospitals or small labs.

Details

A cyclotron is a type of particle accelerator invented by Ernest Lawrence in 1929–1930 at the University of California, Berkeley, and patented in 1932. A cyclotron accelerates charged particles outwards from the center of a flat cylindrical vacuum chamber along a spiral path. The particles are held to a spiral trajectory by a static magnetic field and accelerated by a rapidly varying electric field. Lawrence was awarded the 1939 Nobel Prize in Physics for this invention.

The cyclotron was the first "cyclical" accelerator. The primary accelerators before the development of the cyclotron were electrostatic accelerators, such as the math–Walton generator and the Van de Graaff generator. In these accelerators, particles would cross an accelerating electric field only once. Thus, the energy gained by the particles was limited by the maximum electrical potential that could be achieved across the accelerating region. This potential was in turn limited by electrostatic breakdown to a few million volts. In a cyclotron, by contrast, the particles encounter the accelerating region many times by following a spiral path, so the output energy can be many times the energy gained in a single accelerating step.

Cyclotrons were the most powerful particle accelerator technology until the 1950s, when they were surpassed by the synchrotron. Nonetheless, they are still widely used to produce particle beams for nuclear medicine and basic research. As of 2020, close to 1,500 cyclotrons were in use worldwide for the production of radionuclides for nuclear medicine and ultimately, for the production of radiopharmaceuticals. In addition, cyclotrons can be used for particle therapy, where particle beams are directly applied to patients.

Usage:

Basic research

For several decades, cyclotrons were the best source of high-energy beams for nuclear physics experiments. With the advent of strong focusing synchrotrons, cyclotrons were supplanted as the accelerators capable of producing the highest energies. However, due to their compactness, and therefore lower expense compared to high-energy synchrotrons, cyclotrons are still used to create beams for research where the primary consideration is not achieving the maximum possible energy. Cyclotron-based nuclear physics experiments are used to measure basic properties of isotopes (particularly short lived radioactive isotopes) including half-life, mass, interaction cross sections, and decay schemes.

Medical uses:

Radioisotope production

Cyclotron beams can be used to bombard other atoms to produce short-lived isotopes with a variety of medical uses, including medical imaging and radiotherapy. Positron and gamma emitting isotopes, such as fluorine-18, carbon-11, and technetium-99m are used for PET and SPECT imaging. While cyclotron produced radioisotopes are widely used for diagnostic purposes, therapeutic uses are still largely in development. Proposed isotopes include astatine-211, palladium-103, rhenium-186, and bromine-77, among others.

Beam therapy

The first suggestion that energetic protons could be an effective treatment method was made by Robert R. Wilson in a paper published in 1946 while he was involved in the design of the Harvard Cyclotron Laboratory.

Beams from cyclotrons can be used in particle therapy 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.

As of 2020, there were approximately 80 facilities worldwide for radiotherapy using beams of protons and heavy ions, consisting of a mixture of cyclotrons and synchrotrons. Cyclotrons are primarily used for proton beams, while synchrotrons are used to produce heavier ions.

PET : Positron Emission Tomography.
SPECT : Single Photon Emission Computed Tomography.

Advantages and limitations

Livingston and Lawrence with the 69 cm (27 in) cyclotron at the Lawrence Radiation Laboratory. The metal arch supports the magnet's core, and the large cylindrical boxes contain the coils of wire that generate the magnetic field. The vacuum chamber containing the "dee" electrodes is in the center between the magnet's poles.
The most obvious advantage of a cyclotron over a linear accelerator is that because the same accelerating gap is used many times, it is both more space efficient and more cost efficient; particles can be brought to higher energies in less space, and with less equipment. The compactness of the cyclotron reduces other costs as well, such as foundations, radiation shielding, and the enclosing building. Cyclotrons have a single electrical driver, which saves both equipment and power costs. Furthermore, cyclotrons are able to produce a continuous beam of particles at the target, so the average power passed from a particle beam into a target is relatively high compared to the pulsed beam of a synchrotron.

However, as discussed above, a constant frequency acceleration method is only possible when the accelerated particles are approximately obeying Newton's laws of motion. If the particles become fast enough that relativistic effects become important, the beam slips out of phase with the oscillating electric field, and cannot receive any additional acceleration. The classical cyclotron (constant field and frequency) is therefore only capable of accelerating particles up to a few percent of the speed of light. Synchro-, isochronous, and other types of cyclotrons can overcome this limitation, with the tradeoff of increased complexity and cost.

An additional limitation of cyclotrons is due to space charge effects – the mutual repulsion of the particles in the beam. As the amount of particles (beam current) in a cyclotron beam is increased, the effects of electrostatic repulsion grow stronger until they disrupt the orbits of neighboring particles. This puts a functional limit on the beam intensity, or the number of particles which can be accelerated at one time, as distinct from their energy.

Additional Information

The cyclotron was one of the earliest types of particle accelerators, and is still used as the first stage of some large multi-stage particle accelerators. It makes use of the magnetic force on a moving charge to bend moving charges into a semicircular path between accelerations by an applied electric field. The applied electric field accelerates electrons between the "dees" of the magnetic field region. The field is reversed at the cyclotron frequency to accelerate the electrons back across the gap.

Linear Accelerator

Gist

A linear accelerator (LINAC) is a device that uses high-frequency electromagnetic waves to accelerate charged particles—primarily electrons—to near the speed of light in a straight line, commonly used for cancer radiotherapy. It generates high-energy X-rays or electron beams that precisely target and destroy tumors, minimizing damage to surrounding healthy tissue.

A medical linear accelerator (LINAC) is the device most commonly used for external beam radiation treatments for patients with cancer. It delivers high-energy x-rays or electrons to the region of the patient's tumor.

Summary

A linear particle accelerator (often shortened to linac) is a type of particle accelerator that accelerates charged subatomic particles or ions to a high speed by subjecting them to a series of oscillating electric potentials along a linear beamline. The principles for such machines were proposed by Gustav Ising in 1924, while the first machine that worked was constructed by Rolf Widerøe in 1928 at the RWTH Aachen University. Linacs have many applications: they generate X-rays and high energy electrons for medicinal purposes in radiation therapy, serve as particle injectors for higher-energy accelerators, and are used directly to achieve the highest kinetic energy for light particles (electrons and positrons) for particle physics.

The design of a linac depends on the type of particle that is being accelerated: electrons, protons or ions. Linacs range in size from a cathode-ray tube (which is a type of linac) to the 3.2-kilometre-long (2.0 mi) linac at the SLAC National Accelerator Laboratory in Menlo Park, California.

Details

Linear accelerators, commonly referred to as linacs, are devices designed to accelerate charged particles along a straight path using radio frequency electromagnetic fields. Unlike electrostatic accelerators, which utilize static electric fields, linacs achieve acceleration through specific designs tailored for different particle types, such as electrons, protons, or heavy ions. The two primary designs of linacs are drift-tube and waveguide systems. In a drift-tube linac, charged particles drift through a series of hollow metal tubes, gaining energy at gaps as the voltage alternates to ensure they are continually accelerated.

Waveguide linacs operate by allowing microwaves to guide particles through a hollow pipe, synchronizing the phase velocity of the waves with the particles' speed to maintain energy gain. Linacs are widely used in research, particularly in nuclear physics, where they serve as sources for particle beams. They also have significant applications in medical fields, particularly for radiation therapy and sterilization of medical instruments, as well as in various industrial processes like food preservation and materials testing. The evolution of linacs from initial concepts in the early 20th century to their modern forms has led to advancements in particle energy and beam intensity, making them invaluable tools in both scientific research and industry.

Linear accelerators are devices that augment the energy of a beam of charged particles. Acceleration occurs in a straight line, using radio frequency electromagnetic fields.

Overview

As the name implies, a linear accelerator (linac) is a device in which charged particles travel in a straight line as they are being accelerated. The term also implies in practice that the acceleration is accomplished by means of radio frequency electromagnetic fields and not by means of electrostatic fields as in an electrostatic accelerator. Details of the design of linacs depend on whether they are intended for accelerating electrons, protons, or heavy ions.

Linacs have two basic design types: the drift-tube design and the waveguide design. A drift tube is a hollow metal cylinder, open at both ends; the particles travel along the axis of the cylinder. While they are inside the cylinder and far from either end, the particles do not experience any appreciable force because the electric and magnetic fields are shielded out by the metal of the cylinder. Since the particles experience no force in this region, they are drifting. The essence of the drift-tube method is that a large number of drift tubes are placed one after the other with short gaps between two consecutive tubes. Each tube is arranged to have either a positive or a negative voltage, with the sign alternating from one tube to the next. When particles arrive at the gap between two successive cylinders, the voltages must be arranged so that the particles are repelled from the cylinder they are leaving and attracted to the cylinder they are approaching. When they cross the gap, the particles gain a certain amount of energy. If nothing more were done, they would lose that energy at the next gap, where the force would be in the opposite direction, and the particles would be decelerated. To prevent this loss, the polarities of all the cylinders are reversed when the particles are near the center of the drift tubes. Thus, at every gap the polarity has been arranged for the particles to gain energy. The oscillation in the polarity of the drift tubes takes place at a radio frequency that is characteristic of microwaves.

The particles do not gain much energy at each gap. Nevertheless, the number of tubes (and therefore the number of gaps) can be made quite large, leading to a high value of the beam energy without the need for especially high voltage. In the upstream end of the accelerator, the particles are moving more slowly than they are at the downstream end after they have been accelerated; therefore, the length of the drift tubes must increase gradually from upstream to downstream, more or less proportional to the square root of the tube number. Each gap supplies the same energy increment, and the velocity varies as the square root of the kinetic energy, for nonrelativistic motion.

The drift tubes are all contained inside a long cylindrical tank that is maintained at a good vacuum so that the beam particles are not lost to collisions with air molecules. Devices called klystrons generate the microwaves outside the tank and conduct them to the inside, where they cause electric and magnetic fields primarily at the gaps between drift tubes, secondarily in the region between the drift tubes and the inside wall of the tank, but essentially not deep in the interior of the drift tubes.

Initially, it may seem logical to run the accelerator so that the particles arrive at the gap when the voltage is at its peak, thereby maximizing the energy imparted to the particles in the beam. If this setup is adopted, however, the beam intensity will suffer, since the particles do not all reach the gap in time to take advantage of the full voltage. As a result, the latecomers will get less than the proper gain in energy, causing them to arrive even later at the next gap. Soon, they will be lost to the beam. It is more efficient--and even essential--to design the system with enough extra voltage so that the particles arrive at the gap before the maximum voltage is reached: The particles that arrive early (the ones with too much energy) will receive less than the nominal amount of force at the gap, and their energy will be closer to the right amount. The latecomers (the ones with too little energy) will get to the gap when the voltage is higher than nominal, so they will pick up enough extra energy to get them to the next gap a little earlier. As a result, far fewer particles will be lost to the beam. This is known as the principle of phase stability.

As the particles cross the gaps, the electric forces acting on them tend not only to accelerate them but also to deflect them away from the axis of the system. The latter effect "defocuses" the beam. There are several possible remedies: One can place a small screen across the end of the drift tube to redirect the electric fields; one can use small, specially shaped pieces of metal that accomplish the same purpose as the screen but allow more particles to pass through; or one can use quadruple magnets to impose external focusing on the beam.

A waveguide linac essentially is a hollow pipe with microwaves flowing through it, with the magnetic fields perpendicular to the axis of the pipe and the electric fields at least partially parallel to the axis to impart acceleration to the charged particles in the beam. For all smooth hollow pipes, however, the phase velocity of microwaves is greater than the speed of light, and the special theory of relativity states that no particle can go faster than the speed of light (c). The particles cannot keep up with the accelerating wave. The solution to this problem is to place metal circles at regular intervals along the inside of the pipe. Each circle has a hole in its center large enough to accommodate the passage of the beam. The resulting structure is called an "iris-loaded waveguide." A wave traveling in a periodic structure can be arranged to have any phase velocity one chooses: greater than c, less than c, zero, or even negative. For linacs, the phase velocity of the microwaves is matched to the velocity of the particles being accelerated. Thus, the particles ride the wave exactly as a surfer rides a water wave in the ocean, picking up energy as the wave progresses. If the particles are slightly ahead of the region of maximum field strength, they will also have phase stability.

An alternate way to understand the working of a waveguide linac is to consider it as a large number of cavity oscillators that are connected loosely to one another by the holes where the beam goes. In each cavity there is a standing wave, with the electric field somewhere along the axis so as to provide acceleration. The timing or phase of the oscillations shifts from one cavity to the next so that it is synchronized for the arrival of the particles.

The spacing of the irises (or, alternately, the cavity sizes) varies more for proton or heavy ion linacs than for electron linacs. For the latter, a small energy gain brings the particles' velocity very close to c, and thereafter, they gain energy in regular amounts, with minimal change in the velocity. In a high-energy electron linac, the energy can be varied simply by changing the number of klystrons that drive the accelerator, since the energy is proportional to the square root of the number of klystrons.

Applications

Linacs are used as sources of particles for research in nuclear physics at low, medium, and high energies. The most spectacular example is the 3-kilometer linac at the Stanford Linear Accelerator Center (SLAC) in California. This machine is an electron linac of the waveguide type. It can also accelerate positrons on the opposite half of the cycle, the half that is useless for electrons. The positrons are made inside the accelerator itself by making some of the accelerated electrons impinge on a target located about a third of the way from the start.

At Los Alamos in New Mexico there is a large proton linac that is used for research in medium-energy nuclear physics. This accelerator is best considered as a sequence of cavity oscillators, where alternate cavities do not contribute to the energy gain, but are necessary to maintain accurate timing. These extra cavities are moved out of the beam line so as to shorten the total length of the accelerator.

Linacs are used not only as stand-alone accelerators but also as preaccelerators for other types of accelerators, such as synchrotrons. Proton synchrotrons work better if the protons are relativistic at the outset. The standard procedure is to use a linac that accepts protons at a modest energy and accelerates them to such a high energy that the synchrotron can handle them more effectively. The large accelerator at the Fermi National Accelerator at Batavia, Illinois, is a proton synchrotron built in the form of a ring that is 1 kilometer in radius. The main ring cannot operate effectively with protons direct from the ion source. Instead, the protons from the ion source are accelerated by an electrostatic accelerator of the math-Walton type and then injected into a proton linac, which accelerates them to an energy such that they can be sent to a small synchrotron, after which they go into the main ring. The linac is one indispensable link in a chain of accelerators.

Linacs are also used in pure research as a postaccelerator to an electrostatic accelerator.

Linacs have been built for this purpose with superconducting resonant cavities, using either niobium-copper or lead-plated copper for the walls of the resonator.

The high beam intensity of linear accelerators has made them valuable for medical use, both in radiation therapy and in sterilization of numerous types of small objects used in medicine. For example, surgical thread has to be free of microbes that could cause disease in a patient who has undergone surgery. Radiation from an electron linac can be an effective replacement for either heat or chemicals, the two older methods of sterilization.

The sterilizing properties of radiation have led to varied industrial applications of linacs. They can be used to treat waste water to kill harmful organisms. They can also be used for preservation of food. For example, even a small dose of radiation will prevent the sprouting of onions, garlic, or potatoes at a cost far less than that of the energy needed to refrigerate these food items. At higher levels, the radiation can destroy insects, fungi, or microbes that cause wastage of food.

Other industrial applications exist that do not involve sterilization, such as using radiation to study wear on the surface of metal parts. Many examples of this type of application come from parts of automobile engines: piston rings, cylinder walls, and cams are all metal parts that experience wear from rubbing, in spite of lubrication with oil. It is important to know how the wear takes place and how fast. A linac can provide the radiation needed to perform such studies.

Another class of examples involves irradiation of plastics to induce polymerization and cross-linkage. For example, a plastic object may be formed by pressing the plastic into the required shape. The plastic may "remember" its original form, however, and revert to that rather than keep its new shape. If the plastic is irradiated by the beam from a linac not long after it is pressed, the structure of the polymer tends to be modified so that the plastic "forgets" its original shape.

Additional Information

A linear accelerator is a type of particle accelerator (q.v.) that imparts a series of relatively small increases in energy to subatomic particles as they pass through a sequence of alternating electric fields set up in a linear structure. The small accelerations add together to give the particles a greater energy than could be achieved by the voltage used in one section alone.

In 1924 Gustaf Ising, a Swedish physicist, proposed accelerating particles using alternating electric fields, with “drift tubes” positioned at appropriate intervals to shield the particles during the half-cycle when the field is in the wrong direction for acceleration. Four years later, the Norwegian engineer Rolf Wideröe built the first machine of this kind, successfully accelerating potassium ions to an energy of 50,000 electron volts (50 kiloelectron volts).

Linear machines for accelerating lighter particles, such as protons and electrons, awaited the advent of powerful radio-frequency oscillators, which were developed for radar during World War II. Proton linacs typically operate at frequencies of about 200 megahertz (MHz), while the accelerating force in electron linacs is provided by an electromagnetic field with a microwave frequency of about 3,000 MHz.

The proton linac, designed by the American physicist Luis Alvarez in 1946, is a more efficient variant of Wideröe’s structure. In this accelerator, electric fields are set up as standing waves within a cylindrical metal “resonant cavity,” with drift tubes suspended along the central axis. The largest proton linac is at the Clinton P. Anderson Meson Physics Facility in Los Alamos, N.M., U.S.; it is 875 m (2,870 feet) long and accelerates protons to 800 million electron volts (800 megaelectron volts). For much of its length, this machine utilizes a structural variation, known as the side-coupled cavity accelerator, in which acceleration occurs in on-axis cells that are coupled together by cavities mounted to their sides. These coupling cavities serve to stabilize the performance of the accelerator against changes in the resonant frequencies of the accelerating cells.

Electron linacs utilize traveling waves rather than standing waves. Because of their small mass, electrons travel at close to the speed of light at energies as low as 5 megaelectron volts. They can therefore travel along the linac with the accelerating wave, in effect riding the crest of the wave and thus always experiencing an accelerating field. The world’s longest electron linac is the 3.2-kilometre (2-mile) machine at the Stanford (University) Linear Accelerator Center, Menlo Park, Calif., U.S.; it can accelerate electrons to 50 billion electron volts (50 gigaelectron volts). Much smaller linacs, both proton and electron types, have important practical applications in medicine and in industry.

Q: How did Reese eat her ice cream?
A: Witherspoon.
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Q: What's the difference between a pizza and my pizza jokes?
A: My pizza jokes can't be topped!
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Q: How do you know your close to a Frito Lay factory?
A: Because of the chips and dip in the road.
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Q: What are the 4 major food groups?
A: Pizza, Coffee, Chocolate and Men.
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Q: Why do you always bring a bag of chips to a party?
A: In queso emergency.
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