First circular accelerator that was discovered was cyclotron by Ernest Lawrence in 1940. But this accelerator couldn’t be used for accelerating electrons because these particles acquire a very high speed up to the order of relativistic speed. Hence, its mass starts changing its value by:

$m$ = $\frac{m_{o}}{\sqrt{(1–(\frac{v^{2}}{c^{2}})}}$

Due to the variation in mass , time period and frequency will also start changing their value and hence it is not able to accelerate the particle.Its concept is originated from* Rolf Widroe* who failed in the discovery of induction accelerator because of the lack of traverse focusing. It was the first machine to produce high energy electrons.

Betatron is actually a transformer. It has a torus shaped vacuum tube which acts as the secondary coil of the transformer. An alternating current is applied in the primary coil of the transformer. This accelerates the electrons in the vacuum tube around a circular path. Betatron works under constant electric field and variable magnetic field.**Some of the Applications of Betatron are:**

**Betatron exhibits some limitations as follows:**

- Provides high energy beam electrons of about 300 MeV.
- It can be used as a source of X rays and gamma rays if electron beam is made to direct on a metal plate.
- The X rays produced with the help of betatron can be used in industrial and medical fields.
- High energy electrons can be used in particle physics.
- Possible solar flare mechanism.

The maximum energy range imparted by it to electrons is limited by the strength of the magnetic field because of the saturation of iron and by the size of magnetic core practically.

Synchrotrons overcome these limitations of Betatron.

The oscillations of particles about their stable or equilibrium orbits in all circular accelerators are called **Betatron Oscillations**. These are the stable oscillations about the equilibrium orbit in the horizontal and vertical planes. These are the transverse oscillations of particles in a cyclic accelerator about the equilibrium orbit. Hill’s Equation describes this type of traverse motion by :

$\frac{d^{2}x}{ds^{2}}$ + $K(s)x$ = $0$

Where $K$ is restoring force.

The number of oscillations/turn is Q_{x} or Q_{y}. ( this is known as Betatron Tune). To restore the oscillation back towards the equilibrium, restoring force is required. This restoring force is provided by focusing the components in the magnetic field which bends the particle back towards the equilibrium orbit. If the restoring force (K) is constant in “s” then this is just Simple Harmonic Motion. s is just longitudinal displacement around the ring or torus tube.

In synchrotron accelerators of modern (strong focusing) design there are several cycles of betatron oscillation per revolution of beam particles.

Betatron is a cyclic particle accelerator and is used to accelerate electrons. Operation of **Betatron Acceleration** is explained as follows:

Betatron is a type of transformer where a ring of electrons act as secondary coil. This secondary coil or ring is a vacuum tube in torus shape. An alternating current is applied in the primary coil of the transformer which in turn accelerates the electrons present in the vacuum tube. This makes the electrons to move in the circle inside the tube. They move in the same way as the current is induced in the secondary coil by primary coil as per Faraday’s law.

- The magnetic field which is used to make the electrons follow the circular path is also responsible for accelerating them.
- The magnets should be properly designed such that the average field strength at the orbit radius is equal to half of the average field strength linking the orbit.

$B_{strength}$ = $\frac{Ḃ}{2}$

Where,

$B$ strength is average field strength at the orbit radius.

$Ḃ$ is average field strength linking the orbit.

$\phi$ = $2\pi r^{2}Ḃ$

Here, r is the radius of electron orbit

$B$ is the magnetic field at r.

$\phi$ is the flux inside the are enclosed by orbit of electron.

The magnetic field at the stable orbit should be half the average field over its circular cross section as follows:

$B$ = $\frac{\phi}{2\pi r^{2}}$

Above condition is known as** Wideroe Condition.**

If the magnetic field increases, changing flux links the loop of electrons. This induces electromagnetic field which further accelerates the electrons. As the electrons gain more speed they require a larger magnetic field to move at a constant radius. This magnetic field is provided by the increasing field. The effect is proportional; therefore the field is always strong enough to keep the electrons in same orbit radius.

The magnetic field is changed by passing an alternating current to the primary coils. This makes occurrence of particle acceleration in the first quarter of the voltage sine wave’s cycle. The last quarter of the cycle also has a changing field that could accelerate the particles (electrons) but it is in the wrong direction for them to move in the correct circle. The target is bombarded with pulses of particles at the frequency of the ac supply.

When the magnetic field is at its highest value, the particles gain maximum energy. The formula used for the cyclotron does not work for betatron because the electron will have relativistic nature. However, if total energy is much higher than the rest energy then $E$ = $pc$ is a good approximation. As electrons move in circular orbit, the centripetal force is equal to the Lorentz force:

$\frac{mv^{2}}{r}$ = $q v B_{orbit}$

Maximum momentum would be: $P$ = $r q B_{orbit}$

Where,

$P$ is maximum momentum

$r$ is stable orbit radius

$q$ is charge of electron

$v$ is velocity of electron

$B$ orbit is magnetic field at stable orbit

Therefore,

Energy $E$ = $r q c B_{orbit}$

**Betatron acceleration can be used for the following:**

- Provides high energy beam of electrons of about 300 M eV.
- The X rays produced with the help of Betatron can be used in industrial and medical fields.
- It can be used as a source of X rays and gamma rays if electron beam is made to direct on a metal plate.

Now there exist many new electron accelerators. Some of the best electron accelerators are:

Large electron positron collider, 8$\times$104 M eV.

International linear collider, which can provide electrons of high energy of the order 106 MeV

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