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1. Lect 3:Acceleration by rf Fields
Fayoum University
Faculty of Science
Physics Department
Prepared and Presented by
Dr. Mohammed Attia Mahmoud
- PhD from Fayoum University, Egypt and Antwerp university Belgium
- Member in Egyptian network for High Energy Physics
- Researcher in CMS experiment in CERN, Geneva

2. A. LINAC
B: RFQ
C: Cyclotron
Cynchrocyclotron
D: Microtron
E: Synchrotrons, weak and strong focusing
1.5 Colliders and Storage Rings
1.6 Synchrotron Radiation Storage Rings
II Layout and Components of Accelerators
II. 1 Acceleration Cavities
II. 1 Acceleration Cavities
II.3 Other Important Components
Outlines

3. LINAC
It is difficult to attain very high voltage in a single acceleration gap. It would be more
economical to make the charged particles pass through the acceleration gap many times.
This concept leads to many different rf accelerators,1which can ay betenjan
A. LINAC
In 1925 G. Ising pointed out that particle acceleration can be achieved by using an
alternating radio-frequency field.
 In 1928 R. Wiederoe reported the first working rf accelerator, using a 1-MHz, 25-kV
oscillator to produce 50-kV potassium ions
 In 1931 D.H. Sloan and E.O. Lawrence built a linear accelerator using a 10-MHz, 45 kV
oscillator to produce 1.26 MV Hg+
ion.

4. it consists of a series of metal drift tubes arranged along the beam axis and connected, with
alternating polarity, to a radiofrequency (RF) supply. The supply delivers a high-frequency
alternating voltage of the form U(t) = Umax sinωt.
During a half period the voltage applied to the first drift tube acts to accelerate the particles
leaving the ion source. The particles reach the first drift tube with a velocity v±. They then pass
through this drift tube, which acts as a Faraday cage and shields them from external fields.
Meanwhile the direction of the RF field is reversed without the particles feeling any effect.
When they reach the gap between the first and second drift tubes, they again undergo an
acceleration. This process is repeated for each of the drift tubes. After the i-th drift tube the
particles of charge q have reached an energy

5. From the last eq., the energy is again proportional to the number of stages i
traversed by the particles. The important point, however, is that the largest voltage
in the entire system is never greater than Umax.
During the acceleration the velocity increases monotonically, but the frequency of the
alternating voltage must remain constant, in order to keep the costs of the already very
expensive RF power supply to within reasonable limits. This means that the size of the gaps
between the drift tubes must increase. In the ith drift tube the velocity Vi is reached, which
for a particle of mass m corresponds to an energy
Assuming that, the RF voltage moves through exactly half a period Ƭrf/2 as the particle travels
through one drift section. This immediately fixes the required separation between the ith and
(i + l)th gaps to be
c
v
where i
i
/
, 

The spacing of the accelerating gaps between the drift tubes must thus increase in
proportion with .
i

6. In high-frequency linear accelerator there is still one other problem to solve. The energy
transferred to the particles depends critically on the voltage Umax and the nominal phase Ψo
When a very large number of stages are used, a small deviation from the nominal voltage
Umax means that the particle velocity no longer matches the design velocity fixed by the
length of the drift sections, so that the particles undergo a phase shift relative to the RF
voltage. The synchronization of the particle motion and the RF field is then lost. A mechanism
is thus required to automatically bring the particles back to the nominal phase in the event of
any deviation.
How TO SOLVE IT?
The key principle is not to use the phase Ψo =π/2 and hence
the peak voltage Umax to accelerate the particles, but instead
to use a value Ψo < π/2 . The effective accelerating voltage is
then Ueff < Umax Let us assume that a particle has gained too
much energy in the preceding stage and so is travelling
faster than an ideal particle and hence arrives earlier. It sees
an average RF phase Ψ=Ψo- ΔΨand is accelerated by a
voltage
The particle therefore gains less energy and slows down again until it returns to the nominal
velocity. In practice all particles oscillate about the nominal phase Ψo. This principle of phase
focusing is of crucial importance in the design of all accelerators using RF voltage.

7. The cyclotron
The first circular accelerator to be developed according to this principle was the cyclotron,
proposed by E.O. Lawrence at the University of California in 1930 [13]. A year later Livingston
succeeded in demonstrating the operation of such a machine experimentally. In 1932 they
together built the first cyclotron suitable for experiments, with a peak energy of 1.2 MeV.

9. To make the particles follow a circular path the cyclotron uses an iron magnet which produces
a homogeneous field with a strength of B ≈ 2 T between its two round poles. The particles
circulate in a plane between the poles. The magnetic field then has only one component,
perpendicular to these axes.
We assume that the motion is confined to the x-y plane. The particle momentum then has the
form

10. we finally obtain the required equation of motion
with solutions
The particles thus follow a circular orbit between the poles with a revolution frequency
cyclotron frequency
Notice that ωz does not depend on the particle velocity at all. This is because as
the energy increases, the orbit radius and hence the circumference along which the
particles travel increase in proportion.

11. At higher energies the cyclotron frequency decreases in inverse proportion to the increasing
particle mass m(E). If the frequency of the RF supply is decreased accordingly, much higher
energies can be reached. This principle is employed in the synchrocyclotron. It turns out that
the high frequency is only ever optimal for particles in a limited energy range, and so the beam
may only be accelerated in short pulses or bunches. Hence the beam intensity is lower.
A more effective method is adopted in the isocyclotron, in which the radial magnetic field
is increased in such a way that the cyclotron frequency remains constant, namely
where r(E) is the orbit radius, E is the energy of a particle and q is its charge. However,
this approach has the problem that the beam becomes defocused by the changing
magnetic field. To resolve this, isocyclotrons use magnets with rather complex pole
shapes, which compensate for this loss of focusing with so-called 'edge-focusing'.
Isocyclosynchrotrons can reach energies of over 600 MeV.