Theory of proton synchrotron

Theory of proton synchrotron

Theory of proton synchrotron

In this particular article Theory of Proton Synchrotron, we are going to discuss its construction, working and its theory in detail.

Proton synchrotron

Proton-synchrotron increases the energy of protons. Proton-synchrotron built in Brookhaven National Laboratory increases the energy of protons up to 3000 MeV, whereas proton-synchrotron built in Dubna(Russia) increases the energy of protons up to 10 BeV (10000 Me V). This machine is also known as cosmotron because of particles of such energies can only be found in energetic cosmic rays. The principle of proton-synchrotron essentially the same as that of electron-synchrotron but due to proton much heavier (about 2000 times) than that of an electron, following changes are incorporated in electron-synchrotron.

Theory of Proton Synchrotron

(1) A proton is much heavier than an electron, therefore the velocity of the proton cannot reach up to 0.98c (equivalent to the velocity of electron of energy 2Me V), until the energy of protons becomes more than 4000 Me V. Thus angular frequency of the moving proton increases very rapidly during the acceleration, therefore to maintain the synchronization with the proton moving a stable orbit much change in the frequency of RF electric field will have to be done.

(2) In this accelerator, the injection method of particles of betatron cannot be used. Because of a very heavy magnetic deflector is required. Thus a beam of low energy (1-10 Me V) protons obtained from a linear accelerator or Van de Graaf generator is injected into the doughnut-shaped vacuum chamber when a magnetic field is initiated to increase.


This accelerator consists of a doughnut-shaped vacuum tube known as a race track. The race track is made up of four quadrant arc sections of constant radius connected by four linear sections shown in Fig. These linear sections are used for injecting, accelerating and ejecting the protons. The four quadrant arc sections are placed in between C-shaped electromagnets in such way that the magnetic field produced by an electromagnet is normal the plane of doughnut chamber.

Theory of Proton synchrotron

The protons of energy 1-10 Me V obtained from the linear accelerator or Van de Graaf generator. That is injected tangentially into one of the linear sections. To synchronize the motion of the protons and for providing energy to the protons a very high-frequency electric field is applied in one of the linear sections used as resonant cavity R.


The pre-accelerated protons are injected tangentially into the doughnut-shaped tube from the linear accelerator. Under the influence of a magnetic field in quadrant sections, the protons are made to circulate in the form of a circular arc of constant radius. In each revolution, protons are accelerated once when they pass through the resonant cavity R connected to the RF oscillator. The magnetic field strength and the frequency of the oscillator are simultaneously increased in such a way that the photons travel in a circular path of constant radius. And arrive always at the resonant cavity when an applied voltage is in right phase for acceleration. Thus phase stability is maintained in it. To maintained synchronization, the frequency of the electric field is varied from 250kc/s to 10mc/s. In this condition, protons gain the energy of about 800 eV in each revolution.

Protons stay in the machine for about 1 second. In this time protons make (c/2πrο) revolution along a circular path of radius r0, where c is the speed of light. In the same time, the magnetic field increases up to the maximum level as shown in fig. And then magnetic fields return to the same initial strength at the same time. After keeping the machine at rest for some time again protons are injected into the race track. In this way, high energy protons are obtained in the form of pulses the proton synchrotron.


The angular frequency of revolution of a proton in a circular path of radius r0 is given by the relation


frequency f=ω/π=eB/2πm

or                    f=eBc²/2πmc²=eBc²/2πE

where total energy of the proton


where the k=kinetic energy of a proton and mο in a rest mass of the proton

∴                                                   f=eBc²/2π(K+mοc²)

since race track consists of four quadrants of radius r₀. And four linear sections each of length L, therefore the frequency of revolution of protons decreases in the ratio of a length of the race track. Thus

f=eBc²/2π(K+mοc²).2πο/(2πο+4L)                               ….(1)

From relativistic mass-energy relation,


Squaring it,

or   K²+2m²οc²K+m²ο=p²c²+m²οc²                                                                                    …(2)

Due to the action of magnetic field B, the proton moves in a circular path of radius rο. Hence the relation between  B and proton momentum is given by


∴                               Berοc=√K(K+2mοc²)

or                             B=√K(K+2mοc²)/ecrο                                                                           …(3)

Relation (3) shows how the magnetic field B increases with the increases of kinetic energy K of the protons as it circulates in the orbit.

Consequently, Substituting the equation (3) in equation (1),


or                                                                                    f=[√K(K+2mοc²)/(K+mοc²).c/(2πrο+4L)                                …(4)

In a proton-synchrotron, the applied RF accelerating voltage must to in phase with the circulation frequency f of the proton of stable orbit rο. Hence the frequency of applied RF accelerating voltage must vary in the same manner In which f given by equation (4) varies with proton kinetic energy K.

Theory of Proton synchrotron


In this article Theory of Proton Synchrotron, we have discussed everything about Proton synchrotron. It’s working, its construction and theory.

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