An Introduction to Radio Frequency Acceleration Concepts

A. Methods of Acceleration

This chapter will deal with the concept of Radio Frequency electromagnetic fields and their use as it pertains to particle acceleration. This chapter will present an overview of RF acceleration including definitions of terms commonly used, a generalized view of RF amplification systems, and finally the considerations when transferring beam from one accelerator to another. But first, a little E+M background.

B. Why use RF?

First of all, it should be pointed out that one can not accelerate particles using a magnetic field. By looking at the above force equation and focusing on the magnetic contribution, it can be seen that if one were to take a charged particle at rest and place it in a magnetic field, the particle will experience no force because v is zero, hence the magnetic contribution to the force is zero. Even for a charged particle that is moving, the force created by moving through a static magnetic field is perpendicular to the direction of motion. If the field is in the X-plane, and the particle is traveling in the Z-plane, the force will be in the Y-plane. The equation for the potential energy gained by a particle acted upon by a force is given by:

One can see that for the charged particle example above, that with the force in the Y-plane and the particle traveling in the Z-plane, the dot-product will be zero yielding the energy gained by the particle is zero because the force and the direction of motion are perpendicular. This leaves only an electric field to use to accelerate a charged particle.

Granted, one could use gravity to accelerate charged particles. As an example, to gain 750 keV of potential energy (the amount of energy gained by H- ions when they leave the Pre-accelerator) an H- ion would have to fall :

U = 750 keV 1eV = 1.60219 x10-12 ergs

U = mgh where m = 1.672x10-24g + 2*9.1x10-28 g,

h ~ 5x10¹² cm. This is hardly feasible.

Indeed, to accelerate charged particles, a RF system is really not a necessity. Take the common cathode ray tube (or computer monitor). It has an electron gun which produces a cloud of electrons. There is a high DC potential between this gun (the cathode) and a plate (the anode) near the TV screen, so the electrons are drawn from the electron gun and accelerate towards the TV screen. In fact, this sort of system is the basic principle behind the accelerating column in the Pre-acc. The dome is charged to a potential of -750 kV, and the H- ions accelerate from this negative potential towards the wall of the column which is at ground potential.

For all of the other accelerators at Fermilab, a sinusoidal Radio Frequency system is used for the acceleration. The reason that an alternating electric field is used is that the cavities become resonating structures if fed with a sinusoidal varying electric field. The cavities will resonate at a specific design frequency and any noise or other frequencies will not resonate in the cavity. The beam thus sees only the desired frequency, and not the noise.

The RF voltage produced is applied to an copper cavity with an RF gap in it.

The electric field is induced across this gap and works much the same way a parallel plate capacitor in what the charged particle sees. Each time the particle traverses this gap, it sees a potential difference, and hence is accelerated over this short distance. This effect is multiplied over many traversals of the gap to result in the desired particle energy.

The production of the RF voltage is actually accomplished by producing a RF magnetic field that induces the electric field in the cavity. In most RF cavities there is an input from the RF amplifying system (which we will treat as a black box at this point). This coaxial (two conductor) input to the cavity may look somewhat strange to one who is not familiar with RF systems; it appears as a short in the form of a huge copper bar going from the inner conductor of the coaxial cable to the outer conductor. The time-varying current flowing through this bar produces a large time-varying magnetic field which in turn induces a large time-varying electric field. It is this field that produces the potential across the accelerating gap in the RF cavity.

Now the RF cavities appear to the RF as an R-L-C circuit. The resistance comes from the resistance in the copper walls, the capacitance from the fact that change on the wall of the cavity will induce charge on the center conductor just like a capacitor. The inductance comes from the fact that as current is induced to flow along the walls of the cavity, this produces an axial magnetic field in the same way as an inductor. The capacitance and the inductance of the cavity affect how efficiently the transfer of power between the RF amplifying system (our black box) and the cavity. The most efficient transfer of power would occur when the impedance appears as a simple resistor to the RF amplifying system. An analogy with simple harmonic oscillators from mechanics may be helpful at this point. If you have a mass on a spring, there is a certain frequency of the applied force to the mass which results in the greatest oscillations. This frequency is known as the resonant frequency. In the RF cavities, the situation is slightly different. There is a specific frequency that one wishes to drive the cavity, the goal is to have the cavity resonate at that frequency. Using the mechanics example, this is akin to having a specific spring and driving force and one wants to vary the mass to get a resonant system. In the RF system, one way to affect the resonance of the cavity is to change its volume. This approach is used in Linac and in the Tevatron. The Linac changes the volume by moving a lug or piston in or out to keep the cavity 'in tune.' The Tevatron uses the temperature of the cavity to control the volume. This type of resonance frequency control works best when the RF frequency applied to the cavity is fixed or varies over a very small range.

In Booster and Main Ring, the resonant frequency is controlled by inducing an impedance so that the sum of the induced impedance and the cavity impedance results in the cancellation of the inductive and the capacitive components. The cavities have huge inductors built into the cavity. By having the inductors tuned to cancel the capacitance of the cavity, an near pure resistance can be achieved resulting in an efficient transfer of power.

C. Buckets, Bunches, Batches and all that.

This section will examine the effect of the electric field on particles. Consider an accelerator at a fixed energy (Main Ring sitting at 8 GeV) and an ensemble of protons of energy 8 GeV distributed around the circumference of Main Ring. For the purpose of this discussion, Main Ring will have only one RF cavity. Consider a particle with energy 8 GeV in the Main Ring. From Chapter 2, the velocity of a proton with energy 8 GeV is 98.1792% of the speed of light. For a particle to traverse the circumference of the Main ring, it will take:

Circumference of Main Ring = 2*p*1000m

speed of light = 2.997 *108 m/sec

Time for one trip :   

Frequency of revolutions    

When beam is captured by an RF system, it is contained in what is known as an RF bucket. Since the RF cavity is a resonating structure at a specific RF frequency, there will be standing waves generated within this cavity. These standing wave 'pockets' are the RF buckets. Although this RF structure is present only in the RF cavity, it is helpful to think of this structure existing around the circumference of the accelerator. These buckets do not have to contain beam. If there is RF power being put into the RF cavity, but there is no beam, there still exist RF buckets. If the RF bucket contains beam, then the beam within the bucket is referred to as a bunch. There is another term which is used with regards to beam in Booster, Main Ring and the Tevatron. When Booster circumference is filled with beam (which is usually the case) and the RF is applied, this produces 84 bunches of beam (84 buckets with particles in each bucket). This is referred to a batch. We can inject this batch of beam into Main Ring and then accelerate it, or we can put additional batches of beam from Booster into Main Ring before accelerating. This is known as multi-batch injection, and is used primarily when we are running the Fixed Target program.

To re-iterate, There is an RF structure called a bucket where particles can be placed in order to accelerate them. If a bucket has particles in it, it is referred to a bunch; 84 bunches equals one Booster batch.

There is another term one will hear in the discussion of RF systems. This number is known as the harmonic number. Essentially the harmonic number describes the number of RF buckets in an accelerator. For example the harmonic number of the Booster is 84. This means that there can be a maximum of 84 bunches accelerated in Booster in one acceleration cycle. There are 84 buckets in Booster. Main Ring and the Tevatron have a harmonic number of 1113, so there can be a maximum of 1113 bunches of particles put into either Main Ring or the Tevatron. The antiproton source has many RF systems with various harmonic numbers. Below is a summary of the different harmonic numbers of the various accelerators.

D. Generalized RF amplification system.

This section will describe a 'typical' RF amplification system. Although there are subtle differences between the RF amplification systems used at Fermilab, it will be helpful for the new operator to realize that there is a general layout of the RF system, and the understanding of each system is just the understanding of the specific components used to increase the RF power.

The ultimate goal of the RF system is providing power to be transferred to the beam to increase its energy. The Low Level (LLRF) system is responsible for providing the RF frequency used for the acceleration, as well as taking care of the proper phasing of this RF so that it is 'in time' with the beam. The LLRF system can be thought of as a simple oscillator. This oscillator may be of a fixed frequency, as in the case of Linac, or it may be a frequency that increases as the beam energy increases. A variable or swept frequency is usually generated by a Voltage-Controlled-Oscillator (VCO).

Why is Linac a fixed frequency whereas most of the other accelerators use a varying RF frequency? To answer this, consider what the electric field is doing in the RF cavity when power is applied.

When RF is applied to an infinitely long cavity, the particle accelerates while the electric field is in the positive direction. As the electric field begins to change direction, the accelerated particle would experience a decelerating force. The particle motion would be oscillatory around a fixed point with no net displacement, or increase in energy.

What would be helpful would be shielding the particle from the negative electric field. This is in fact what is done. As the particle is accelerated it is driven towards a grounded tube. During the negative electric field, the particle is shielded from it and just drifts through the grounded tube. As it re-emerges from the 'drift tube' it experiences a positive electric field and hence receives an accelerating force.

 

As the particle picks up more and more velocity, the time it takes to pass through the drift tube shortens. If the frequency is fixed, the particle will start arriving at the end of the drift tube when there is still a negative electric field. There are two options available to the designers of RF cavities. The RF frequency can be increased so that by the time the particle leaves the drift tube, there is a positive electric field to accelerate the particle. Another alternative is that the drift tubes can be made longer giving the particle a longer drift space to traverse while the electric field is in the negative direction. This last methods is best suited to RF cavities where beam makes one pass through the structure such as a Linac cavity. The former method is used in circular accelerators where changing the drift space during acceleration would be very difficult.

 

Once the LLRF is provided, it goes next to a system where it's power will be increased (on the order of kW). This is done with a power amplifier, modulator, and an anode supply.

The anode supply is a big raw power supply that provides a raw high voltage to the modulator. The modulator takes this raw high voltage and regulates based on a programmed wave form typically called the anode curve. The modulator supplies this regulated high voltage to the power amplifier. The power amplifier takes the regulated high voltage from the Modulator and combines it with the LLRF signal to produce a modulated high power high frequency wave form to the RF cavity. Another component of the typical RF system is what is called a bias supply. This supply provides current to control the inductance of the RF cavity, thereby making it appear as a pure resistance to the power amplifier in order to create the most favorable conditions for effective power transfer. Below is a table describing the connection between the ideal RF system above with the RF systems you will likely see at Fermilab.

As portrayed in the drawing above, there is a feedback system that applies corrections to the RF that is being pumped into the cavity. A feedback system is a system that measures a parameter of some system and compares it to a reference. The difference between the measured value and the reference becomes an 'error signal'. This error signal is applied to the system as a correction in order to reduce the error signal. In the case of the RF system, the feedback systems can be a phase feedback system or an energy feedback system, or both. We will discuss the energy feedback system first.

For an energy feedback system, there is usually a desired radial position that it is desirable to keep the beam at in order to keep the beam centered in the beampipe. The actual position of the beam is determined by using a Beam Position Monitor (BPM). Subtracting this actual beam position signal from our desired (reference) position will result in an error signal. This is applied to the LLRF system to change the phase of the RF in order to change the energy of the beam to center it in the beampipe.

In order to talk about the phase feedback system it will be necessary to digress a little and talk about something known as the synchronous phase.

A particle in a circular machine that is synchronized to the RF frequency such that on each consecutive revolution it passes through the RF gap when the RF is at the same phase is said to be a synchronous particle. For coasting beam this means the particle will get enough acceleration on one pass to make up for the energy lost during one revolution. On each consecutive pass it will undergo the same acceleration since it passes through the gap during the same RF phase.

A particle that is not a synchronous particle will receive more or less acceleration than a synchronous particle, when it passes through the RF gap. On the following pass the particle will have the wrong energy, and will arrive at the gap at the wrong time. The particle will be accelerated appropriately, but will have the wrong phase during its next pass through the RF gap. This is an oscillation about the correct RF phase and is called a synchrotron oscillation. Particles undergoing synchrotron oscillations about an ideal particle will either have a large enough energy/phase error that they no longer are effected by the RF and are lost, or they will continue with their synchronous oscillations until the oscillations are damped out. Synchrotron oscillations will have the effect of spreading the particles throughout the stable region of the RF bucket. This increases the emmitances in the longitudinal direction. For the Main Ring at 8 GeV typical synchrotron oscillations are about 500 Hertz.

The phase feedback loop is used to control the phase of the RF with respect to the phase of the beam. Whereas the energy feedback loop can be thought of affecting the position of beam radially, the phase feedback system will also affect the position if the beam longitudinally (along the axis of travel) is not centered in the RF bucket. The diagram below shows an RF bucket and the five possible energy/phase errors a bunch of beam could have. The bunch in the middle is at the design energy and at the synchronous phase. The phase feedback loop measures the phase of the beam with respect to the synchronous phase, and delays or advances the phase of the RF in the cavity so that it matches the phase of the beam.