Subatomic Physics

26 Accelerators
particles the push at the right time. Equation (2.17) then shows that the applied rf
must increase with increasing energy up to the point where the particles are fully
relativistic so that pc = E. The magnetic field also must increase:
ω = kΩ = kc pc
R E
kc pc
−→ ; B = . (2.19)
R |q|ρ
If these two conditions are satisfied, then the particles are properly accelerated.
The procedure is as follows: A burst of particles of energy Ei is injected at the time
t = 0. The magnetic field and the rf are then increased from their initial values
Bi and ωi to final values Bf and ωf, always maintaining the relations (2.19). The
energy of the bunch of particles is increased during this process from the injection
energy Ei to the final energy Ef . The time required for bringing the particles up to
the final energy depends on the size of the machine; for very big machines, a pulse
per sec is about par.
Equation (2.19) shows another feature of these big accelerators: particles cannot
be accelerated from start to the final energy in one ring. The range over which the
rf and the magnetic field would have to vary is too big. The particles are therefore
preaccelerated in smaller machines and then injected. Consider, for instance, the
1000 GeV synchrotron at FNAL: The enormous dimensions of the entire enterprise
are evident from Photo 4. (10)
Synchrotrons can accelerate protons or electrons. Electron synchrotrons share
one property with other circular electron accelerators: they are an intense source
of short-wavelength light. The origin of synchrotron radiation can be explained
on the basis of classical electrodynamics. Maxwell’s equations predict that any
accelerated charged particle radiates. A particle that is forced to remain in a circular
orbit is continuously accelerated in the direction toward the center, and it emits
electromagnetic radiation. The power radiated by a particle with charge e moving
with velocity v = βc on a circular path of radius R is given by (11)
P = 2e2 c
3R 2
β 4
(1 − β2 . (2.20)
) 2
The velocity of a relativistic particle is close to c; with Eqs. (1.6) and (1.9) and with
β ≈ 1, Eq. (2.20) becomes
P ≈ 2e2c 3R2 γ4 = 2e2c 3R2 �
E
mc2 �4 . (2.21)
The time T for one revolution is given by Eq. (2.16), and the energy lost in one
revolution is
10J. R. Sanford, Annu. Rev. Nucl. Sci. 26, 151 (1976); H. T. Edwards, Annu. Rev. Nucl.
Part. Sci. 35; 605 (1985).
11Jackson, Eq. (14.31).