Cyclotron
A cyclotron is a device developed by Lawrence and Livingstone by which the positively charge particles like proton deuteron etc can be accelerated.
Principle. The working of the cyclotron is based on the fact that a positively charged particle can be accelerated to a sufficiently high energy with the gulp of smaller values of oscillating electric fields by making it to cross the same electric field time and ageing with the used of strong magnetic field.
Construction it consist of two D - shaped hollow evacuated metal chambers D1 and D2 called the dees. These dees are placed horizontally with their diametric edges parallel and slightly separated from each other. The dees are connected to thigh frequency oscillator which can produce a potential difference of the order of 10 volts at frequency = 10 Hz. The two dees are enclosed in an evacuated steel box and are well insulated from it. The box is paved in a strong magnetic field produced by two pole pieces of strong electromagnets N, S the magnetic field is perpendicular to the plane of the dees P is a pave of ionic source or positively charged particle.
Working and theory the positive ion to be accelerated is produced at p suppose, at that instant D1 is at negative potential and D2 is at positive potential therefore the ion will be accelerated towards D1. On reaching inside D1 the ion will be in a field free space hence it mobs with a constant speed in D1 say v. but due to perpendicular magnetic field of strength B the ion will describe a circular path of radius r (say) in Dr. given by Bq v = mv2 / r
Where m and q are the mass and charge of the ion.
∴ r = m v / Bq
Time taken by ion to describe a semicircular path is given by
t = πr /v = πm / Bq = B π / (q / m) = a constant.
thus this time is independent of both the speed of the ion and radius of the circular path. In case the time during which the positive ion describes a semicircular path is equal to the time during which half cycle of electric oscillator is completed then as the ion arrives in the gap between the two dees the polarity of the dees is reversed D1 becomes positive and D2 negative. Therefore, the positive inn is accelerated towardsD2 and it enters D2 with greater speed which remains constant in D2. The ions will describe a semicircular path of greater radius due to perpendicular magnetic field and again will arrive in gap between the two dees exactly at the instant the polarity of the two dees is reversed. Thus the positive ion will go on accelerating every time it comes into the gap between the dees and will go on describing circular path of greater and greater radius and finally acquires a sufficiently high energy. The accelerated ion can be removed out of the dees from window W, by applying the electric field
across the deflecting plates E and F.
Maximum energy of positive ion
let v0, r0 = maximum velocity and maximum radish of the circular path followed by the positive ion in cyclotron.
Then mv20 / r0 = Bq v0 or v0 = Bq r0 / m
Max K.E = 1 / 2 m (Bq r0 / m )2 = B2 q2 r20 / 2m
∴ Max KE = 1 / 2 mv20
Cyclotron Frequency
If T is the time period of oscillating electric field then T = 2t = 2 πm/ Bq
The cyclotron frequency is given by
V = 1 / T = Bq / 2 πm
It is also known as magnetic resonance frequency.
They cyclotron angular frequency is given by
We = 2 πv = Bq / m
Limitations of cyclotron
When a positive ion is accelerated by the cyclotrons it moves with greater and greater speed as the speed of the ion becomes comparable with that of light the mass of the ion increase according to the relation.
M = m0 / 1 - v2 / c2
Where mo = the rest mass of the ion,
M = the mass of the ion while moving with velocity v,
C = velocity of light.
Now the time taken by the ion to describe semicircular path is
T = π m/ Bq = π / Bq m0/ 1 - v2 / c2
It shows that as v increase t increases. It means the positive ion will take longer time to describe semicircular path then the time for half cycle of oscillating electric field. A a result of ti, the ion will not arrive in the gap between the two dees exactly at the instant, the polarity of the two dees is reversed and hence will not be accelerated further. Therefore the ion cannot move with a speed beyond a certain limit in a cyclotron.
Cyclotron is suitable only for accelerating heavy particles like proton deuteron particle etc. electrons cannot be accelerated by the cyclotron because the mass of the electron is small and a small increase in energy of the electron makes the electrons move with a very high speed. As a result of it the electrons go quickly out of step with oscillating electric field.
The uncharged particles (neutrons cannot be accelerated by cyclotron.
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