Longitudinal Focusing & The Gamma Transition The Synchrotron Longitudinal Focusing & The Gamma Transition Inspired by Oliver Bruning and the Particle Accelerator Group

Longitudinal Focusing As a particle beam is introduced into a synchrotron and begins to circulate, the particles in the beam gain and loss energy. Longitudinal focusing is an intrinsic design of the accelerator and works in such a way as to keep the particles in the beam together.

Formulas That Apply The mass of the particle is represented by the equation, m =  mo where  (gamma) = 1/√ 1 – v 2/c 2. v is the velocity of the particle and m0 is its rest mass. This is shown by the following formulas. •  = (q/ mo ) (B/), and • r = (mo /q)(/B)v According to the formula, r is directly proportional to  and v. Since E= mc2 & p = mc , an increase in  yields an increase in the energy and momentum of the particle. th

When charged particles accelerate, they radiate energy When charged particles accelerate, they radiate energy. As they do so, they deviate from the beam. Longitudinal focusing maintains a well-defined energy spread of the beam in a self-stabilizing manner. Let us follow the path of a particle in the beam with energy that is less than the reference energy of the beam. If the particle continues on this track, it will soon be out of sync with the beam and will cause a spread in the energy of the beam. In order to remedy this situation, the beam passes through a cavity with an alternating voltage applied to it, called an RF cavity. If the beam arrives in sync with the voltage drop across the cavity, it experiences a zero voltage drop and continues on without any change in energy. However, if a particle has lost energy it arrives in the RF cavity, at t = t0 - t, because the angular frequency has increased. The particle receives an acceleration due to the positive voltage drop. Refer to Figure 1 below.

V Voltage vs time graph for RF Cavity t t0 - t t0 t0 + t Figure 1

The particle’s energy is increased and it arrives in sync with the beam in the next turn or so. This process is continuous as the particle continues to gain and loose energy/momentum If the particle’s energy is much greater than that of the beam, then the angular frequency of the particle decreases and it arrives in the RF cavity at a time t = t0 + t. The voltage is now in the negative phase of its cycle and the particle experiences deceleration. This process continues until it arrives in sync with the beam..

Gamma Transition There is a dividing point along the energy spectrum of the synchrotron. The behavior of the beam is different in the regions above, at and below this energy divide. See Figure 2 below.

Figure 2 Graph of Gamma Transition E  > t  = t  < t B Energy of Beam vs Magnetic Field

Case 1:  < t A beam of particles with energy below t will behave in the following way. If a particle’s energy is higher than the reference energy of the beam, its frequency of revolution decreases, and the particle arrives in the RF cavity when the reference voltage is negative, at t = t0 -t. It will experience a deceleration that will slow it down and put it more in phase with the bundle. Likewise if it is traveling to slow with too little energy, the particle will arrive in the RF cavity later at, t =t0 +t,. It will experience an acceleration and increase its energy Refer to Figure 1 above

In this region,  increases with . Case 2:  > t In this region,  increases with . When the beam operates in this region, there is a 180 degree shift in the voltage verse time graph. This gives the opposite results of what one would expect to happen in the  < t region. In the case of a beam that has too much energy, its radius of orbit increases and also its speed/momentum; however, the frequency of revolution increases as well, unlike what happens to a particle at  < t,, and the particle arrives at t = t0 -t, to complete the cycle earlier. The RF curve is now in the negative voltage cycle and the particle experiences an acceleration until it finds itself once again in sync with the RF.

In the case a beam that has too little energy its radius of orbit decreases as does its speed/momentum; however, the frequency of revolution decreases as well, unlike what happens to a particle at  < t,, and the particle arrives at t = t0 + t, arriving later to complete the cycle. The RF curve is now in the positive voltage cycle and the particle experiences a acceleration until it finds itself once again in sync with the RF.

Voltage vs time graph for RF Cavity-  > t t0 - t t0 + t t t0 Figure 3

Case 3:  = t There is a point, called gamma transition, t , where the particles stop oscillating between less and more energy. The particles continues to orbit the synchrotron but do not adjust for energy differences to insure longitudinal focusing. Thus the beams gradually spreads in energy and becomes more incoherent.   The beam cannot stay in this region at t. It must move above t into the more relativistic region of operation of the synchrotron in order to produce particles with higher energies. However the behavior of the particles does not seem to be continuous across the t...We know that it is continuous and one has to be very careful and closely monitor the beam when moving across t..