Non-relativistic
The centripetal force is provided by the axle alluring acreage B, and the force on a atom travelling in a alluring acreage (which causes it to be angularly displaced, i.e. spiral) is according to Bqv. So,
\frac{mv^2}{r} = Bqv
(Where m is the accumulation of the particle, q is its charge, B the alluring acreage strength, v is its dispatch and r is the ambit of its path.)
The dispatch at which the particles access the cyclotron due to a abeyant difference, V.
v = \sqrt{\frac{2Vq}{m}}
Therefore,
\frac{v}{r} = \frac{Bq}{m}
v/r is according to angular velocity, ω, so
\omega = \frac{Bq}{m}
And back the angular abundance is
ω = 2πf
Therefore,
f = \frac{Bq}{2\pi m}
The abundance of the active voltage is artlessly the changed of this abundance so that the atom crosses amid the dees at the aforementioned point in the voltage cycle.
f_c = \frac{Bq}{2\pi m}
A brace of "dee" electrodes with loops of coolant pipes on their apparent at the Lawrence Hall of Science. The atom avenue point may be apparent at the top of the high dee, area the ambition would be positioned
This shows that for a atom of connected mass, the abundance does not depend aloft the ambit of the particle's orbit. As the axle spirals out, its abundance does not decrease, and it accept to abide to accelerate, as it is travelling added ambit in the aforementioned time.
edit Relativistic
As particles access the dispatch of light, they access added mass, acute modifications to the abundance (synchrocyclotron), or the alluring acreage during the dispatch (isochronous cyclotron).
edit Synchrocyclotron
The ambit of curvature for a atom affective relativistically in a changeless alluring acreage is
r = \frac{\gamma m_0 v}{q B}
where
\gamma=\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}} is the Lorentz factor
m0 is the blow accumulation of the particle.
Note that in high-energy abstracts energy, E, and momentum, p, are acclimated rather than velocity, and both abstinent in units of energy. In that case one should use the substitution,
\frac{p}{E} = v
area this is in Natural units
The relativistic cyclotron abundance is
f=f_c\sqrt{1-\left(\frac{v}{c}\right)^2},
where
fc is the classical frequency, accustomed above, of a answerable atom with velocity
v ambit in a alluring field.
The blow accumulation of an electron is 511 keV/c2, so the abundance alteration is 1% for a alluring exhaustion tube with a 5.11 keV/c2 absolute accepted accelerating voltage. The proton accumulation is about two thousand times the electron mass, so the 1% alteration activity is about 9 MeV, which is acceptable to abet nuclear reactions.
edit Accessory cyclotron
An another to the synchrocyclotron is the accessory cyclotron, which has a alluring acreage that increases with radius, rather than with time. Recalling that
r = \frac{\gamma m_0 v}{q B},
one can accept B to be proportional to the Lorentz factor, B = γB0. This after-effects in the relation
r = \frac{m_0 v}{q B_0}
which afresh alone depends on the dispatch v, like in the non-relativistic case. The axle de-focusing aftereffect of this adorable acreage acclivity is compensated by ridges on the allurement faces which alter the acreage azimuthally as well. This allows particles to be accelerated continuously, on every aeon of the radio abundance (RF), rather than in bursts as in a lot of added accelerator types. This assumption that alternating acreage gradients accept a net absorption aftereffect is alleged able focusing. It was obscurely accepted apparently continued afore it was put into practice.
The centripetal force is provided by the axle alluring acreage B, and the force on a atom travelling in a alluring acreage (which causes it to be angularly displaced, i.e. spiral) is according to Bqv. So,
\frac{mv^2}{r} = Bqv
(Where m is the accumulation of the particle, q is its charge, B the alluring acreage strength, v is its dispatch and r is the ambit of its path.)
The dispatch at which the particles access the cyclotron due to a abeyant difference, V.
v = \sqrt{\frac{2Vq}{m}}
Therefore,
\frac{v}{r} = \frac{Bq}{m}
v/r is according to angular velocity, ω, so
\omega = \frac{Bq}{m}
And back the angular abundance is
ω = 2πf
Therefore,
f = \frac{Bq}{2\pi m}
The abundance of the active voltage is artlessly the changed of this abundance so that the atom crosses amid the dees at the aforementioned point in the voltage cycle.
f_c = \frac{Bq}{2\pi m}
A brace of "dee" electrodes with loops of coolant pipes on their apparent at the Lawrence Hall of Science. The atom avenue point may be apparent at the top of the high dee, area the ambition would be positioned
This shows that for a atom of connected mass, the abundance does not depend aloft the ambit of the particle's orbit. As the axle spirals out, its abundance does not decrease, and it accept to abide to accelerate, as it is travelling added ambit in the aforementioned time.
edit Relativistic
As particles access the dispatch of light, they access added mass, acute modifications to the abundance (synchrocyclotron), or the alluring acreage during the dispatch (isochronous cyclotron).
edit Synchrocyclotron
The ambit of curvature for a atom affective relativistically in a changeless alluring acreage is
r = \frac{\gamma m_0 v}{q B}
where
\gamma=\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}} is the Lorentz factor
m0 is the blow accumulation of the particle.
Note that in high-energy abstracts energy, E, and momentum, p, are acclimated rather than velocity, and both abstinent in units of energy. In that case one should use the substitution,
\frac{p}{E} = v
area this is in Natural units
The relativistic cyclotron abundance is
f=f_c\sqrt{1-\left(\frac{v}{c}\right)^2},
where
fc is the classical frequency, accustomed above, of a answerable atom with velocity
v ambit in a alluring field.
The blow accumulation of an electron is 511 keV/c2, so the abundance alteration is 1% for a alluring exhaustion tube with a 5.11 keV/c2 absolute accepted accelerating voltage. The proton accumulation is about two thousand times the electron mass, so the 1% alteration activity is about 9 MeV, which is acceptable to abet nuclear reactions.
edit Accessory cyclotron
An another to the synchrocyclotron is the accessory cyclotron, which has a alluring acreage that increases with radius, rather than with time. Recalling that
r = \frac{\gamma m_0 v}{q B},
one can accept B to be proportional to the Lorentz factor, B = γB0. This after-effects in the relation
r = \frac{m_0 v}{q B_0}
which afresh alone depends on the dispatch v, like in the non-relativistic case. The axle de-focusing aftereffect of this adorable acreage acclivity is compensated by ridges on the allurement faces which alter the acreage azimuthally as well. This allows particles to be accelerated continuously, on every aeon of the radio abundance (RF), rather than in bursts as in a lot of added accelerator types. This assumption that alternating acreage gradients accept a net absorption aftereffect is alleged able focusing. It was obscurely accepted apparently continued afore it was put into practice.
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