Weak focusing I. mv r. Only on the reference orbit is zero

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Sheena Phoebe Gibson
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1 Weak focusing I y x F x mv r 2 evb y Only on the reference orbit is zero r R x R(1 x/ R) B y R By x By B0y x B0y 1 x B0 y x R

2 Weak focusing (II) Field index F x mv R 2 x R 1 n

3 Betatron frequency 2 Fx mx v x x x 0 x 1n 0 1n R The oscillation frequency is called betatron frequency my evb x To obtain vertical stability we need a vertical restore force -> Horizontal component of B B x B y y x 0 Fy Cy B B B x R R y 0y 0y Bx dy n dy n y Bx Cy

4 Vertical focalization n>0 The field decrease as the radius increase But 1-n>0?? 2 0 y y y 0 y n x v n n R 0 1 n y

5 1-n Centrifugal Force and Magnetic Force must be balanced at least for a value of orbit radius If B y does not fall faster than 1/r (1-n>0) there is always a radius in which the curves cross and the two forces are balanced

6 Weak focusing The principle of weak focusing has one serious drawback: Since the betatron oscillation wavelength is larger than the circumference of the machine one gets large deviations from the orbit if the circumference is large. The magnet apertures must be very big. The apertures can be drastically reduced if one applies strong focusing (n much larger than 1). This is impossible in a machine which has a guide and focusing field independent of the azimuthal angle, since in that case the condition 0 < n < 1 has to hold, as we have just shown. It is, however, possible if we split up the machine into a series of magnetic sectors in which in alternating order the magnetic field increases strongly with increasing radius ( n << 1) or decreases strongly with increasing radius.

7 Strong focusing The overall effect is focusing!!

8 The magnets for big accelerators strong focusing combined function dipole magnets with alternating strong radial field gradient no possibility for fine-tuning separated function homogeneous dipole magnets for bending quadrupole magnets for focusing

9 Summary Betatron focusing and guiding fields Condition for stability Wideroe 1/2 Betatron oscillation around the reference orbits Weak focusing Limit of weak focusing Strong focusing Separated function magnets

10 The cyclotron

11 Cyclotron animation

12 Cyclotron frequency Cyclotron frequency Not dependent on the particle velocity Valid only for non relativistic case The precision of the B field must be very high in the order of 0.1% B in the order of Tesla, voltage in the order of hundreds of kev

13 Exercise Which RF frequency applied on dees is needed to accelerate a proton in a uniform, 1 Tesla magnetic field?

14 The first cyclotron First cyclotron realized in cm: 80 kev 28 cm: 1 MeV 69 cm: ~5 MeV

15 Shims

16 Kinetic energy Ion with atomic number A and charge ne

17 Numbers Let s assume B=1.5 T, R=0.5 m In a superconductive cyclotron B=4.6 T, R=0.7 m, so the energy could be around 500 MeV/nucleon

18 Exercise Considering E c in the order of 100 kev To achieve 100 MeV we need about 1000 revolution Which is the relative increment of the orbit path length for every revolution?

19 Relativistic particles Cyclotrons increase the energy of the particles by the same amount of energy at each turn. At low energy, the particles cross the gap at fixed frequency. At higher energy when relativistic corrections start to matter, the frequency at which they cross the gaps starts to decrease (the particles travel at the same speed ~c but follow a longer path). This can be addressed by varying the drive frequency but only all particles in the cyclotron are nearly at the same energy. The classic cyclotron is limited to about 20 MeV mainly due to the relativistic mass increasing

20 Synchro Cyclotron Veksler and McMillan showed, independently of each other, that by adjusting the frequency of the applied voltage to the decreasing frequency of the rotating protons, it was possible to accelerate the protons to several hundred MeV. Whereas the cyclotron can accelerate a stream of particles, the synchro-cyclotron can only accelerate one bunch of particles Alternatively, one could vary the magnetic field to keep the revolution frequency constant. This is an isochronous cyclotron.

21 Isocyclotron Varying the height of the iron between the magnet poles the magnetic field changes for different azimuthal angles The average value increases with the radius Additional focusing is provided by the pole edges Energy up to 600 MeV

22 Exercise Find the radial dependence of the magnetic field B z (r) in an isocyclotron with angular frequency ω z

23 Cyclotrons today Mainly used in the medicine field for the production of short live radionuclides But also for ion beam therapy Still used in nuclear physics

24 Ion treatment With proton or ion treatment the does localization is much more effective This is a consequence of the Bragg peak

Historical developments. of particle acceleration

Historical developments. of particle acceleration Historical developments of particle acceleration Y.Papaphilippou N. Catalan-Lasheras USPAS, Cornell University, Ithaca, NY 20 th June 1 st July 2005 1 Outline Principles of Linear Acceleration Electrostatic