Cyclotron: Definition, Principle, Construction, Working, and Uses

A cyclotron is a particle accelerator used to accelerate charged particles or ions to high energy values. It was invented in the year \(1934\) by scientists E.O Lawrence and M.S. Livingston. Until that time, much less was known about the atomic nucleus, and scientists worldwide were working on designing experiments or instruments that could help them understand the nucleus of an atom. It all began with the invention of the first linear accelerator in \(1928\) by Leo Szilard, which inspired Lawrence to develop the cyclotron to investigate the stricture of the atomic nucleus.

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What are Particle Accelerators?

Particle accelerators are machines that are used to accelerate subatomic particles like electrons or protons to very high speeds by applying a combination of electric and magnetic fields. These particles are contained properly into well-defined beams. These accelerated particles are either made to smash into a fixed target or made to collide against other particles. These collisions are very useful for research in particle physics, studying the world of infinitely small and the generation of X-rays or gamma rays.

What is a Cyclotron?

A cyclotron is a particle accelerator in which an electromagnetic field is applied to increase the energy of the subatomic particles. Under the action of the alternating electric field and a static magnetic field, the charged particles or ions follow a spiral path. The fields applied in a cyclotron are perpendicular to each other. That is why these are called crossed fields, when a charged particle is inserted within a cyclotron, ensuring that its direction is perpendicular to the direction of the static magnetic field. The subatomic particle rotates each time due to the magnetic field, while the applied electric field causes it to accelerate. A high-frequency alternating voltage generates the electric field. The frequency of the cyclotron matches the frequency of the alternating voltage, and it is usually kept constant.

A cyclotron is designed on the concept that the frequency of revolution of a charged particle in a magnetic field does not depend on the energy of the charged particle.

Principle of Cyclotron

The working of a cyclotron can be explained by the principle – when a moving charged particle enters a magnetic field normally, it experiences a magnetic Lorentz force that causes the particle to move in a circular path. Its operation can be understood from the fact that the time required for a single revolution of a charged particle is independent of the radius of its orbit or its speed.

Construction of Cyclotron

A cyclotron has three main parts:
1. A high-frequency source to generate alternating voltage.
2. A huge electromagnet to provide a uniform magnetic field.
3. A pair of small hollow half-cylinders composed of highly conductive material called Dees.

In a cyclotron, a hollow metal cylinder is divided into two hollow sections – \({D_1}\) and \({D_2}.\) Since their shape closely resemble the shape of the letter \(D,\) these are called Dees. These dees are very small, around \(21\)′′ in diameter, and are separated by a small distance with their straight sides facing each other. At the centre, in the gap between the dees, a source of ions is placed. The dees are kept between the poles of the strong electromagnet such that the magnetic field acts perpendicular to the plane of the dees. A high potential difference of the order \({10^5}\,{\rm{V}}\) exists between the faces of the dees that are provided by an oscillator connected across their terminals. This high-frequency oscillator works on the frequency of the order \(10 – 15\) megacycles per second, and it changes the charge on each dee many million times in one second.
The whole set-up is placed within an evacuated chamber such that there exists a pressure of \({10^{ – 6}}\,{\rm{mm}}\) of \({\rm{Hg}}\) within the two dees.

Force on a Charged Particle in a Uniform Magnetic Field

In a cyclotron, a combination of electric and magnetic fields interact with the charged particle. The force due to magnetic field \(B\) acting on a charge \(q\) that is moving with a velocity \(v,\) can be given as:
\(F = q\left( {v \times B} \right)\)
Or, \(F = qvB\,\sin \,\theta \)
If the charge is moving perpendicular to the magnetic field, then,
\(F = qvB\,\sin \,\left( {{{90}^{\rm{o}}}} \right)\)
Thus, \(F = qvB\)
The direction of this force is perpendicular to the direction of the velocity of the charged particle and the applied magnetic field.

Frequency of Rotation of the Charged Particle

The charged particle, under the effect of the magnetic field, rotates in a circular path. Any particle in circular motion experiences a centripetal force acting along its centre. Thus, the centripetal force acting on a particle of mass \(m,\) moving with velocity \(v\) in a circle \(pf\) radius \(r\) can be given as:
\({F_c} = \frac{{m{v^2}}}{r}\)
The magnetic field provides this centripetal force. Thus,
\({F_c} = F\)
\(\frac{{m{v^2}}}{r} = qvB\)
\(v = \frac{{qBr}}{m}\)
We know that the angular frequency of cyclotron, \(\omega = \frac{v}{r}\)
Thus, \(\omega = \frac{{qBr}}{{mr}} = \frac{{qB}}{m}\)
The frequency of cyclotron can be given as \(f = \frac{\omega }{{2\pi }}\)
Substituting the values from above,
\(f = \frac{{qB}}{{2\pi m}}\)
This is the cyclotron frequency.

Working of a Cyclotron

Let us suppose that the alternating potential applied across the dees, at any instant, is such that it makes \({D_1}\) positive and \({D_2}\) negative. Now a positive ion is emitted from the source \(S\) kept within the two dees. This positive charge will be attracted towards the negatively charged \({D_2}.\) Within the metallic dee, the ion will not be affected by the electric field. A uniform magnetic field is applied across the dees at the right angle to the plane of the dees. This magnetic field will act on the positive ion and force it to move in a circular path. If the magnitude of the magnetic field is \(B,\) the charge on the ion is \(q,\) and its mass is \(m,\) and \(v\) is the speed of the particle, then, the radius of its motion,
\(r = \frac{{mv}}{{qB}}\)

The frequency of the oscillator is adjusted in a way such that as soon as the particle reaches the edge of \({D_2},\) the polarity of potential difference on the two dees gets reversed, thus, \({D_1}\) becomes negative and \({D_2}\) becomes positive. As the ion reaches in the middle, it gets attracted towards, \({D_1}.\) The electric field will accelerate the positive ion; thus, its speed will increase. The ion will now move in a circular path at a constant speed which will be higher than its speed in \({D_2}\) and the radius of its path will, therefore, be larger in \({D_1}\) as compared to \({D_2}.\)

As the ion comes out of \({D_1},\) the polarity of the electric field is again reversed so that it will accelerate the ion, giving it additional energy. In this manner, the ion will continue to move between the two dees, with its radius increasing after each half cycle.

If \(T\) is the time period of the revolution of the ion, that means it covers the semicircular path in any one of the dees in time \(\frac{T}{2}.\) The time taken by the ion to complete one cycle can be given as:
\(T = \frac{1}{{{v_c}}} = \frac{{2\pi m}}{{qB}}\)

Therefore, by varying the strength of the magnetic field, the period of the revolution of the ion can be varied. This is done to ensure that the time taken by ion to move across the semi-circle is the same as the time taken to change the polarity of potential on the dees. Thus, ion always enters the electric field in the correct phase.

These ions can be accelerated to high values of energies by this method. When the radius of the path traversed by the ions becomes equivalent to the circumference of the dees, these ions are deflected from their circular path and moved towards a fixed target by a magnetic field.

The entire setup is evacuated to ensure minimum collisions between air molecules and the ion.

The Kinetic Energy of Charged Particles

We know, in a cyclotron, the frequency of the applied voltage is adjusted so that polarities of the dees are reversed as soon as the ions complete half a revolution. Thus, if \({v_a}\) and \({v_c}\) be the frequency of applied voltage and cyclotron.
Then, \({v_a} = {v_c}.\)
This is called resonance condition. The supply voltage’s phase is adjusted to ensure that when the positive ion reaches the edge of \({D_1},\,{D_2}\) will be at the lower potential. Thus, ions will be accelerated across the gap between the dees.

If a charge \(q\) is accelerated through the dees and a potential \(V\) is applied across the dees, then the kinetic energy of charge will increase by \(qV\) every time the charge crosses the gap between the dees. The radius of the path of the charged ions increases as it repeatedly goes through the dees. When the radius of their path equals the radius of the dees, the charged particles have gained sufficient energy. They are deflected by a magnetic field and leave the cyclotron by an exit slit. If \(R\) is the radius of the trajectory followed by the charged particle, then its velocity,
\(v = \frac{{qBR}}{m}\)
Thus, the kinetic energy of the particle,
\(K.E. = \frac{1}{2}m{v^2}\)
Or,
\(K.E. = \frac{1}{2}\frac{{{q^2}{B^2}{R^2}}}{m}\)

Uses of a Cyclotron

1. Cyclotrons are useful in nuclear reactions for bombarding atomic nuclei for the production of X-rays.
2. Cyclotrons have been proven useful in radiation therapy, especially in the treatment of cancer.
3. Cyclotrons are useful in studying nuclear transmutations.

Limitations of a Cyclotron

1. Cyclotrons can accelerate ions, but the speed achieved using cyclotrons are very small compared to the speed of light.
2. Cyclotrons can not be used to accelerate neutral particles like neutrons.
3. Cyclotron can not be used to accelerate electrons because electrons have low mass, and hence their speed increases very quickly. It is almost impossible to maintain the resonance condition for the electrons.

Summary

A cyclotron is a particle accelerator in which an electromagnetic field is applied to increase the energy of the subatomic particles. Under the action of the alternating electric field and a static magnetic field, the charged particles or ions follow a spiral path. The working of a cyclotron can be explained by the principle – when a moving charged particle enters a magnetic field normally, it experiences a magnetic Lorentz force that causes the particle to move in a circular path.

In a cyclotron, a hollow metal cylinder is divided into two hollow sections – \({D_1}\) and \({D_2}.\) The dees are kept between the poles of the strong electromagnet such that the magnetic field acts perpendicular to the plane of the dees. A high potential difference of the order \({10^5}\,{\rm{V}}\) exists between the faces of the dees that are provided by an oscillator connected across their terminals. The alternating electric field accelerates ions to high energies, and when their energy increases to a sufficiently high value, the ions exit the dees from a slit.

Frequently Asked Questions (FAQs) on Cyclotron

Q.1. What is a cyclotron?
Ans: A cyclotron is a particle accelerator used to accelerate charged particles or ions to high energy values.

Q.2. What are the three main components of a cyclotron?
Ans: Three main components of a cyclotron are:
1. Large electromagnets.
2. Highly conductive cylinder divided into two parts called Dees.
3. A high-frequency oscillator.

Q.3. Why do we place the cyclotron set up in an evacuated chamber?
Ans: We place the set-up in an evacuated chamber to minimize the charged ions’ energy loss due to collision between ions and air molecules.

Q.4. What is the resonance condition of cyclotron?
Ans: When the frequency of applied voltage is equal to the cyclotron frequency, the cyclotron is said to be in resonance condition.

Q.5. How can we vary the cyclotron frequency?
Ans: The formula for cyclotron frequency can be given as \(f = \frac{{qB}}{{2\pi m}}.\) Thus, for a given charge, cyclotron frequency can be varied by varying the applied magnetic field.

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