Phase Stability in Particle Accelerators

(in Russian, avtofazirovka), the phenomenon permitting acceleration of electrons, protons, α-particles, and multi-charged ions to high energies (from several MeV to hundreds of GeV) in most charged-particle accelerators. This principle was discovered by the Soviet physicist V. I. Veksler in 1944 and independently by the American physicist E. McMillan in 1945. This phenomenon played a fundamental role in extending the range of attainable energies in cyclic accelerators.

In cyclic accelerators, particles execute orbital motion in a special vacuum chamber in a magnetic field and repeatedly traverse accelerating electrodes. Particle acceleration occurs in response to a high-frequency electric field applied to the accelerating electrodes. Continuous particle acceleration requires that in the moment of acceleration the directions of motion of the particle and the electric field coincide; this calls for a synchronism (resonance) between the particle motion and changes in the electric field. If the amplitude of the potential difference between the electrodes is V₀, then the energy ∆E acquired by a particle of charge e in each transit through the accelerating gap will be ∆E = eV₀ cos ø, where ø is the phase of the electric field at the time of transit of the particle, as counted from the peak value of the field. The phase field ø, at which the particle traverses the accelerating gap, is known as the particle phase, for brevity.

In order for the particle to remain in step with changes in the accelerating field, the particle frequency of revolution ω must be equal to or some multiple of the frequency ω₀ of the electric field: ω₀ = qω, where q is an integer (resonance index). Then the particle will traverse the accelerating electrodes at the same phase value ø and will receive the same energy from the field in each traversal. It will therefore be continually accelerating. This situation takes place in the cyclotron—the only resonance accelerator that existed before the discovery of the phase-stability principle. In a cyclotron, particles move in a constant magnetic field H at a constant frequency of revolution ω = eH/mc (where m is the particle mass and c is the speed of light). Consequently, when the frequency of the accelerating electric field is ω₀ = ω for all particles, exact resonance with the field is observed.

However, the mass m can no longer be assumed constant when a sufficiently high energy is attained: the effect of increasing particle mass with increasing energy then begins to become prominent. The increased mass means a smaller frequency of revolution ω and a breakdown in the resonance between particle motion and accelerating field. Particles cease to acquire energy from the electric field and slip out of the acceleration mode. Therefore, a conventional cyclotron features a limiting energy beyond which further acceleration is impossible. This energy limit is about 20 MeV in the case of protons.

Resonance can be retained by slowly decreasing the frequency ω₀ of the accelerating field in proportion to decreasing ω, by slowly altering the magnetic field strength H in order to compensate the loss of frequency ω, or by combining both procedures.

However, the hundreds and thousands of billions of particles being accelerated simultaneously in the machine have a wide distribution of energies and consequently of masses. This means that particles will have different frequencies of revolution ω. This in turn signifies that it is impossible to achieve an exact resonance with the accelerating field to move this huge assemblage of accelerated particles. This difficulty seemed to be insuperable prior to the discovery of the phase-stability principle.

What Veksler and McMillan showed was that precisely because of this dependence of the frequency of revolution of the particle on the particle energy (or particle mass), which was responsible for the loss of exact synchronism between particle motion and the accelerating field, the field itself will automatically restore the synchronism, averaged over several resonances, for a large number of particles. In other words, when ω depends on the energy, an accelerating field of frequency ω₀ (which may be a slowly varying field) constrains the particles to orbit at average frequencies equal to (or some multiple of) the frequency ω₀ (that is, achieving resonance in the average); the particle phases oscillate and concentrate about some one phase ø₀, which is called the synchronous phase, or equilibrium phase. This phenomenon is known as phase stability.

Phase stability results in the particles on the average orbiting in step with changes in the accelerating field: ω_ɑ&=ω₀. Consider how phase stability is achieved in a cyclic accelerator with a uniform magnetic field, constant in time, at q = 1. The frequency of revolution of particles in that accelerator is inversely proportional to the particle mass and consequently to the total energy of the particles (equal to the sum of the rest energy and the kinetic energy). A synchronous particle (an imaginary particle moving in exact resonance with the accelerating field) will experience acceleration at the same phase ø₀, each time acquiring an energy eV₀ cos ø₀. The synchronous phase ø₀ must be positive—that is, it must ride on the falling side of the accelerating voltage wave—for particle orbital motion to be phase-stable—that is, so that particles with phases ø ≠ ø₀ are not lost from the acceleration mode. Actually, a particle of lower energy whose frequency of revolution ω is greater than ω₀ and which is moving in step with the synchronous particle at some instant will eventually leave the synchronous particle, arriving earlier at the accelerating gap and becoming accelerated at a smaller phase ø₁ < ø₀. Consequently, the particle will require more energy, eV₀ cos ø₁ = ø₀ cos ø, and its frequency will begin to decline, so that the exact resonance ω = ω₀ will occur at some moment. However, this resonance will be only instantaneous, for the particle will be receiving a greater energy from the field than before, and its frequency ω will continue to decline for some time and will become less than the synchronous frequency: ω - ω₀. The particle will then lag behind the synchronous particle and will acquire less energy from the accelerating field than the synchronous particle, so that its frequency will again increase.

A similar process occurs with a particle lagging behind the synchronous particle and arriving somewhat later at the accelerating depth, at a phase ϕ2 = ϕ₀. Such a particle will receive less energy from the field, its frequency will begin to increase, and the particle will overtake the synchronous particle.

Thus, the frequency of revolution of the particles will perform oscillations about the equilibrium value ω₀ which are slower than the frequencies of revolution of the synchronous particle. The particle phases will therefore oscillate above the value ø₀, and the average phase of the particles will be stable: ø_av = ø₀ (hence the name phase stability). The synchronism between the particle motion and the accelerating field will therefore be maintained automatically on the average. Simultaneously, the other characteristics of the particle motion (energy and orbit radius) will also oscillate about their equilibrium values corresponding to those of the synchronous particle. These phase oscillations and the related oscillations in the orbit radius of the particle are known as radial-phase oscillations.

Phase stability also occurs in linear resonance proton accelerators, in which (in contrast to cyclic accelerators) the frequency of transit of successive accelerating gaps (arranged in a straight line) by a particle is directly proportional to the velocity of the particle’s motion—that is, it increases with increasing energy. However, a stable synchronous phase is negative in linear accelerators—it falls on the rising side of the accelerating electric voltage wave. The field increases as the particle crosses the accelerating gap, so that a lagging particle (of phase ø₂ = ø₀) acquires more energy and begins to overtake the synchronous particle, while a leading particle (of phase ø₁ - ø₀) acquires less energy and also begins to move closer to the synchronous particle.

The phase-stability principle exerted a revolutionizing effect on the development of accelerator engineering. A family of accelerators of different types operating on the basis of phase-stability made its appearance: cyclic electronic accelerators (synchrotrons) with energies as high as 7 GeV, cyclic proton accelerators (the phasotrons or synchrocyclotrons, the synchrophasotrons or proton synchrotrons, and others) with energies as high as 75 GeV, cyclic accelerators of variable resonance index q (microtrons), and linear resonant proton accelerators with energies as high as 70 MeV. Phase stability is absent when the frequency of revolution of the particles is independent of their energy (in the case of isochronous cyclotrons); it is also absent in linear accelerators when the speed of particle motion comes close to the speed of light and virtually ceases to be energy-dependent (linear electronic accelerators with energies above 10 MeV).