Betatron

Betatron is a particle accelerator that is used to accelerate beta particles, usually electrons. It works on the principle of electromagnetic induction to accelerate charged particles to high energy particles. They were one of the earliest particle accelerator developed for research and development in the area of particle physics.

In this article, we will learn in detail about betatron, its construction, working principle, advantages, disadvantages, uses and limitations. We will also compare betatron with cyclotron in this article.

Table of Content

What is a Betatron?
Properties of Betatron
Types of Betatrons
Principle of Betatron
Betatron Construction and Working
Difference between Betatron and Cyclotron

What is a Betatron?

Betatron is a circular induction accelerator used for electron acceleration. A betatron, utilized in high-energy physics, propels electrons to relativistic velocities. In other words, we can describe a betatron as a type of particle accelerator that produces high-energy electrons, or positrons. It operates on the principle of electromagnetic induction. It consists of a large toroidal vacuum chamber surrounded by an electromagnet. Alternating current (AC) is passed through the electromagnet, generating a time-varying magnetic field. Consequently, this field induces a voltage in a metal tube within the chamber, accelerating electrons to high energies as they spiral around the chamber due to the Lorentz force.

Properties of Betatron

The properties of betatron are mentioned below:

Energy of a Betatron

The maximum electron kinetic energy achieved by betatrons is about 300 MeV. The energy limit is determined in part by the practical size of pulsed magnets and in part by synchrotron radiation.

Betatron Frequency

The betatron frequency or v value is the frequency of the betatron motion of the circulating beam per one revolution in the ring. One important parameter of particle dynamics in an accelerator is the betatron frequency and its dependence on a particle’s amplitude.

The first mention of the betatron frequency was in the 1941 pioneering work by Kerst and Serber. They defined it as the fractional number of particle oscillations around the orbit per one revolution period in a betatron.

Conditions for Betatron

A betatron acts as a secondary coil of the transformer. It helps to accelerate the electrons only in a vacuum. This process of acceleration can only be conducted within a circular vacuum tube. Betatron is functional under the conditions of the variable magnetic field and constant electric field.

Oscillation in Betatron

The particles undergo oscillatory motion within the Betatron. Electrons move back and forth along their circular path as they gain energy. The oscillations are driven by the alternating magnetic field, which continuously accelerates the particles. The oscillation of the particle is in stable orbits. The motion of the particle is described by Hill’s equation which is given as

d²x/dt² + ω²(t)x=0

Where

x(t) is the unknown function of time.
t is the independent variable (time).
ω(t) is a function of time, representing the frequency or angular frequency of the oscillations.

Types of Betatrons

There are mainly two types of betatrons:

Classic Betatron
Resonant Betatron

Classic Betatron

Classic betatron is also known as a vacuum betatron. It operates by inducing an alternating magnetic field within a vacuum chamber using a large doughnut-shaped magnet and a high-frequency alternating current (AC). As the magnetic field changes, it induces a circular motion in the electrons, causing them to accelerate. The classic betatron was the original design proposed by Donald Kerst and was first demonstrated in 1940.

Resonant Betatron

Resonant betatron is also known as a magnetic induction accelerator. It uses a series of induction coils or magnets arranged along the path of the particle beam. The magnets produce a changing magnetic field, which induces a resonant oscillation in the particles. The resonant betatron can operate at higher energies and with higher efficiency compared to the classic betatron.

Apart from the above types of betatron, they are also classified on the basis of the shape of the vacuum chamber. Based on this, the betatrons are classified as follows:

Circular Betatron
Racetrack Betatron

Circular Betatron

The traditional Betatron design consists of a circular vacuum chamber with a toroidal shape.
Electrons are accelerated along a circular path inside the chamber by the changing magnetic field generated by the surrounding electromagnet.
Circular Betatrons are commonly used in research laboratories and medical facilities for various applications, including medical imaging and radiation therapy.

Racetrack Betatron

Racetrack Betatron has a straight section connected to semi-circular ends, resembling a racetrack or oval shape.
Electrons are accelerated along the straight section before entering the semi-circular ends where they complete their circular path.
Racetrack Betatrons are less common than circular Betatrons but may offer certain advantages in specific applications or research settings.

The above types of Betatrons operate on the same principle of electromagnetic induction to accelerate electrons, but they may differ in their design and application. The choice between circular and racetrack Betatrons depends on factors such as the desired energy output, space constraints, and other factors.

Principle of Betatron

Betatron particle accelerator operates on the following principle:

When an electric current is passed through the magnet, it generates a strong magnetic field inside the chamber.
To accelerate the electrons, a series of alternating current (AC) pulses are sent through a coil located inside the chamber.
These pulses create a rapidly changing magnetic field, which in turn induces an electric field.
The electric field accelerates the electrons in a circular path within the chamber.
As the electrons gain energy, their velocity increases, and they move in larger orbits.
This process continues until the electrons reach their maximum speed or desired energy level.

Betatron Construction and Working

The construction and working of Betatron is discussed below:

Construction of Betatron

A betatron consist of following parts:

Vacuum Chamber: Betatron has a vacuum chamber also called Doughnut Chamber to contain the path of the accelerated particles. It is positioned between two electromagnet poles and are driven by an alternating current(AC), with a frequency of 80Hz to 180 Hz. Vacuum chamber also minimizes collisions between the accelerated particles and air molecules, ensuring efficient acceleration.
Primary Coil: A large coil of wire, known as the primary coil, is wound around the vacuum chamber. The primary coil is connected to an alternating current (AC) power supply. This AC power supply generates the changing magnetic field necessary for particle acceleration through electromagnetic induction.
Injection System: Charged particles, such as electrons, are injected into the vacuum chamber at a specific injection point. The injection system include electron guns or other similar devices to introduce the particles into the accelerating region of the betatron.
Detector Systems: Betatrons also have detector systems to monitor and measure the accelerated particles’ properties, such as energy, trajectory, and intensity.
Power Supply: A power supply is required to provide electrical power to the primary coil for generating the alternating magnetic field. The power supply must be enough to achieve the desired magnetic field strength and frequency.
Cooling System: Betatrons also have cooling system. This is because the primary coil and other components of the betatron may generate heat during operation. Hence, to maintain optimal performance it require cooling system.

Working of Betatron

The working of Betatron is discussed in detail below:

Generation of Alternating Magnetic Field: A large coil of wire is connected to an alternating current (AC) power supply. When the AC power supply is turned on, it generates a alternating magnetic field within the vacuum chamber of the betatron. This changing magnetic field induces an electric field within the vacuum chamber due to electromagnetic induction.
Injection of Charged Particles: Charged particles, such as electrons, are injected into the vacuum chamber at a specific injection point.
Acceleration of Particles: As the charged particles travel along the circular path within the vacuum chamber, they experience an electric field induced by the changing magnetic field. This electric field exerts a force on the particles, accelerating them as they move around the circular path.
Energy Gain: With each pass through the accelerating region of the betatron, the particles gain energy due to the induced electric field. The energy gained by the particles is proportional to the strength and frequency of the magnetic field, as well as the number of revolutions they make around the circular path.
Detection and Analysis: After being accelerated, the particles can be directed towards various experimental apparatus or detectors for analysis.

When the electromagnets are powered and an electron occurs at K (the doughnut tube’s cathode), the magnetic field grows. This growing magnetic field has two effects, which are:

By altering the magnetic flux, which provides the electron with additional energy, induced e.m.f. is created in the electron orbit. Faraday’s law says that e.m.f = –(dΦ)/(dt)
The operation of a magnetic field whose direction is perpendicular to the electron velocity causes a radial force (magnetic force) that maintains the electron’s circular motion. Centripetal force balances the force. This is given as qvB = (mv²)/r

The electron is only held in the tube for T/4 seconds because the particle acceleration only happens during the period when the flux grows from zero to its greatest value. After this, the flux starts to decrease, which causes the electron’s velocity to decrease. The growing field provides the greater magnetic field that the quicker electrons require to maintain their constant radius of motion.

Difference between Betatron and Cyclotron

Betatron and Cyclotron are two commonly used particle accelerators. The difference between betatron and cyclotron is tabulated below:

Betatron	Cyclotron
It is a particular type of particle accelerator modified primarily to accelerate beta particles or electrons.	A cyclotron is a type of particle accelerator that uses a spiral path to accelerate charged particles.
Upto 300 MeV of electron energy is accelerated.	Upto 80 MeV energies are used to accelerate positive ions.
It has expanding magnetic field.	It has constant magnetic field.
Electrons are accelerated as long as the betatron state is sustained.	Ions are accelerated as long as the resonance state is sustained.
Only the first part of each cycle experiences electron acceleration.	At the apex of each half cycle, ions accelerate.
Betatron uses a circular path for accelerating charged particles.	Cyclotron uses a spiral path for accelerating charged particles.
Betatron is modern compared to cyclotron.	Cyclotron is the earliest form of the accelerator
Its path is circular	Its path is semicircular or spiral

Advantages and Disadvantages of Betatron

The advantages and disadvantages of betatron are discussed below:

Advantages of Betatron

The advantages of betatron are:

Betatron has the advantage that it produce full voltage on a secondary coil and then apply that voltage to a high-vacuum x-ray tube.
Betatrons can achieve high energy gains for charged particles in a relatively short distance, making them efficient for certain particle acceleration tasks.
They typically have simpler designs compared to other types of accelerators, leading to easier maintenance and lower downtime for repairs.
Betatrons can operate continuously once they are started, which is advantageous for applications requiring a steady beam of particles.
They are relatively compact compared to other types of particle accelerators like cyclotrons or linear accelerators. This makes them suitable for applications where space is limited.

Disadvantages of Betatron

The disadvantages of betatron are:

Only a circular vacuum tube is capable of conducting the acceleration process.
Only in a vacuum can the electrons be accelerated.
The procedure by which electrons are ejected is complex.
When there is a fluctuating magnetic field and a steady electric field, the betatron can operate.
Power requirements are huge.

Uses of Betatron

Betatrons are used for following applications:

The high energy electrons can be used in the field of particle physics
Betatron provides high energy beam electrons of about 300 MeV.
It is used as a source of X-rays and gamma rays if the electron beam is directed onto a metal plate. It has industrial applications and is used in the medical field.
It is used for Radiography.
They are used in medical applications, particularly in radiation therapy for cancer treatment. They can generate high-energy electron beams that can be focused on tumors with precision, minimizing damage to surrounding healthy tissue.

Limitation of Betatron

Although, Betatrons offer many advantages in terms of compactness and simplicity compared to other particle accelerators, they also have limitations related to energy output, size, beam quality, and operational costs.

Energy Limitation: The maximum energy Betatrons can impart to particles is limited. Due to design constraints and practical considerations, they may not achieve as high energies as other types of particle accelerators such as linear accelerators or synchrotrons.
Size and Complexity: Despite being more compact than linear accelerators, Betatrons still require a large infrastructure, including the electromagnet and vacuum chamber. This can limit their practicality for certain applications where space is limited or where portability is desired.
Fixed Energy Output: Betatrons typically operate at a fixed energy output determined by the design of the electromagnet and vacuum chamber. Unlike synchrotrons, Betatrons cannot easily change their energy output during operation.
Radiation Hazard: Betatrons produce high-energy electron beams, which can pose radiation hazards if not properly shielded and controlled.
Beam Quality: Betatrons may have limitations in beam quality, including beam stability, divergence, and energy spread. These factors can affect the accuracy and precision of experiments or applications using electron beam produced by the them.
Maintenance and Operating Costs: Maintaining and operating Betatrons can be costly due to its complex equipment involved, including the electromagnet, vacuum system, and associated control systems.

Related Articles
Motion of a Charged Particle in a Magnetic Field	Electromagnetic Field
Relativistic Mass Formula	Displacement Current

Betatron Frequently Asked Questions

What is betatron accelerator?

The word betatron derives from the fact that high-energy electrons are often called β-particles. The betatron is a circular induction accelerator used for electron acceleration.

Who Invented the Betatron?

Betatron was invented by Donald W. Kerst in 1940 at the University of Illinois, Urbana-Champaign.

How Does a Betatron Work?

When an electric current is passed through the magnet, it generates a strong magnetic field inside the chamber. To accelerate the electrons, a series of alternating current (AC) pulses are sent through a coil located inside the chamber. These pulses create a rapidly changing magnetic field, which in turn induces an electric field. The electric field accelerates the electrons in a circular path within the chamber. As the electrons gain energy, their velocity increases, and they move in larger orbits. This process continues until the electrons reach their maximum speed or desired energy level.