Issue 1, November 2003

Physical Sciences & Mathematics

Analysis of Electron Cloud Build-Up in High Energy Particle Accelerators Using the Java™ Programming Language

Laura Loiacono
Loyola University
Advisor: Dr. Katherine Harkay
Argonne National Laboratory

The electrons in an electron cloud may interfere with the stored beam in an accelerator, causing a shortened lifetime, an expanded size, and a decrease in the quality of the beam. To increase the performance of an accelerator, accelerator physicists must understand the mechanisms behind the formation, growth, and interaction of electron clouds.

At APS, the electron cloud is dominated by secondary electrons of three types: rediffused electrons, elastically scattered electrons, and true secondary electrons (SEs). Rediffused electrons are not well understood, and do not contribute much to the electron cloud. Elastically scattered electrons are created when an electron from the cloud collides with a chamber wall and is reflected back into the chamber with the same energy. SEs are created when an electron from the cloud collides with the chamber surface, and the energy of the collision is transferred to the atoms in the metal, liberating one or more electrons from the atoms in the metal surface. Since SEs have the greatest probability of forming at collision energies greater than 10 eV, and beam-cloud interactions usually accelerate electrons to energies higher than this, SEs dominate the electron cloud (Redhead et al. 1968).

Experiments at APS and other accelerator facilities have shown that the amplitude of the electron cloud is affected by the properties of the particle beam itself. The particle beam is not actually a continuous beam; rather, the particles (positrons or electrons) are distributed throughout the ring in a number of "bunches," each of which interacts separately with cloud electrons. The time between the passage of one bunch and the arrival of the next bunch at a single location is referred to as the "bunch spacing". Experimental data shows that the amplitude of the electron cloud is a function of bunch spacing (Harkay and Rosenberg 1999).

The amplification in the electron cloud can be explained by classical "beam-induced multipacting," which was first suggested by Grobner (1977). Beam-induced multipacting is a resonance condition resulting from the Coulomb interaction between low-energy cloud electrons and high-energy, charged beam particles. Classical beam-induced multipacting theory suggests that secondary electrons created at the wall receive a momentum kick from the beam and traverse the chamber to collide with the opposite wall in exactly one bunch spacing. The wall collision energy creates new electrons, which receive the same momentum kick at the wall. This repetitive process leads to the rapid growth of the cloud, but it does not explain how bunch spacing affects this growth.

Furman and Heifets have proposed a theory, known as the Furman-Heifets resonance theory, which attempts to explain amplification of the electron cloud using a more general beam-induced multipacting theory. Unlike classical beam-induced multipacting, the Furman-Heifets resonance theory includes the secondary electron energy distribution (2000). According to this theory, we assume an electron is created at the wall of the vacuum chamber with energy between 1 and 3 electron volts (eV) (Figure 1a), the most probable energy range in the emitted SE spectrum (Redhead et al. 1968). Its corresponding momentum (p), given by

p = ( Ex2xmc² )^1/2,

where E is the electron’s energy and mc² is the electron mass in eV, is between 1010 and 1750 eV/c. Then the electron drifts to a distance, a₁, from the beam, where it receives a momentum kick, dp₁, from a passing bunch (Figure 2b). The electron is accelerated across the chamber and collides with the opposite wall, creating one or more secondary electrons with energy assumed to be between 1-3 eV (Figure 1c). The newly created secondary electrons drift to a distance, a₂, which is the same distance as that of a₁, where they receive a momentum kick, dp₂, equal to dp₁, from a passing bunch. This recurring process is called a resonance condition, and can rapidly increase the already large concentration of secondary electrons within the storage chamber. This could explain the amplification of the electron cloud for certain bunch spacings.

If the more general Furman-Heifets resonance condition contributes to the amplification of an electron cloud, then it is important to characterize the conditions under which a resonance forms. In order to examine these conditions, several computer programs were created in the Java programming language (Sun Microsystems 2002). By calculating the momentum kick imparted to electrons of the electron cloud, the resulting distance that electrons travel, and their positions in the chamber at the instant the next bunch passes, the programs can be used to search for those initial beam and electron cloud parameters that give rise to a resonance condition.

Engineering drawings of the vacuum chamber in which the electron cloud forms were used to find mathematical equations defining the shape of the inner surfaces of the chamber (ANL 1991). The chamber is symmetric about the horizontal and vertical axes, and its shape is described by four circular arcs and four line segments. Figure 2 shows a schematic of the chamber cross-section, as well as the algebraic equations used to define it. A two-dimensional coordinate system was used to locate the position of an electron, as well as the length of its "path" within the vacuum chamber.

Figure 2 The shape of the inner surface of a cross-section of the vacuum chamber is defined by algebraic equations for an upper and lower arc, a left and right arc, and four line segments. The points at which the lines join in the upper right quadrant of the coordinate system are given by angles of 63 and 79 degrees measured clockwise from the positive y-axis. The vertical and horizontal half-diameters of the chamber are 2.0853 cm and 4.2341 cm, respectively. (Click to view enlarged image)

It is assumed that the interaction between the beam and electron cloud occurs in a two-dimension cross-section of the vacuum chamber, as shown in Figure 3. It is also assumed that one electron starts a distance a_i away from the beam and at angular position q , measured clockwise from the positive y-axis. The electron’s motion is assumed to be restricted to a line through the center of the chamber at an angle q , which is referred to as the "path". The path of the electron has a corresponding wall-to-wall length, termed pathlength, that is constant for a given angle q . The electron’s distance from the nearest wall is b_i, and its distance from the far wall is termed remainingPath_i. (Figure 3). The electron is given an initial momentum, p_i, and its velocity is assumed to be directed toward the beam. These variables account for the position of the electron along its path after successive interactions with the beam.

Realistically, the cloud electrons interact with each bunch for approximately 40 ps, which corresponds to the root-mean-square (rms) bunch length. However, the cloud-beam interaction can be simplified by using an impulse kick approximation whereby the bunch length is zero and the force felt by the cloud electrons due to a passing bunch is instantaneous. The momentum kick is given by

Δ p = 2m_ecr_eN_b/r x e(r),

where m_e is the electron mass, c is the speed of light, r_e is the electron radius, N_b is the charge per bunch, r is the radial distance from the center of the bunch, and e(r) is a radial dependent function. The effect of the radial dependence is a decrease in the magnitude of the kick for electrons positioned very close to the center of the beam (small r).

Programs SE_B and ESE_B simulate both non-impulse kick and impulse kick interactions. For the purpose of the code, the dynamics of both processes are equivalent. The difference is that for the former it is necessary to integrate the Coulomb force over the bunch length by dividing the bunch into a number of small segments and carrying out the beam-cloud interaction calculations for each segment. The non-impulse kick dynamics simulated in both programs also allow for varying charge distributions. The computer programs calculate interactions for rectangular and Gaussian charge distributions.

For simplicity, only the impulse kick approximation is discussed in this paper; however, the discussion is also pertinent to the non-impulse kick simulation with the bucket spacing and charge per bunch replaced by the bunch length and charge per bunch both divided by the user-selected number of bunch divisions.

The programs SE_B and ESE_B simulate beam-cloud interactions for a positron and electron beam, respectively. A positron beam will attract the negatively charged electrons in the electron cloud; thus, the kick received by an electron will always act to accelerate it in the direction of the beam. Conversely, an electron beam will repel the negatively charged electrons in the electron cloud, and the electrons’ momentum change will always be directed away from the beam. A detailed explanation of positron beam dynamics is presented in this paper. The same dynamics can be applied to electrons if the opposite sign momentum kick is applied.

For a position beam, the cloud electrons are accelerated toward the beam during interactions. If the electrons receive a first kick of magnitude dp₁ eV/c, the kick adds to the electron’s initial momentum to give it a new momentum, p₁ eV/c. With momentum p1 eV/c, the electron travels a distance distTrav₁ in the time tbucket between bunch passes. Tbucket is the same as the time between successive kicks (Appendix A.1).

After receiving a kick, the electron may or may not collide with the chamber wall. Three "cases" provide an overview of the possible scenarios that could result after the first kick. Case 1 (Figure 4) occurs when the distance that the electron travels, distTrav1, in time tbucket is less than its distance from the beam, a_i (Appendix A.2). In other words, the electron does not cross the path of the beam. In this case, it is still traveling toward the beam when the next bunch passes. Case 2 (Figure 5) results when the electron crosses the beam path but does not reach the opposite wall (Appendix A.3). The electron is then traveling away from the beam at the instant the next bunch passes. Finally, Case 3 (Figure 6) occurs when the electron travels across the chamber, collides with the opposite wall, and creates a secondary electron with energy between 1 and 3 eV (1010 and 1750 eV/c). In this case, the time timeUsed that the electron took to travel the distance to the wall may be less than the time between bunches or kicks, tbucket. The newly created secondary electron then has the remaining time, timeRemaining, to travel, or drift, a distance, driftDist, toward the beam before the next bunch passes and gives it a second kick. If in that time the secondary electron collides with the opposite wall, it may produce another secondary electron. If it collides multiple times with the chamber walls, then it will produce multiple electrons (Appendix A.5). (By re-assigning the values of a₁, b₁ and remainingPath₁ (Figure 7) the outcome becomes the mathematical equivalent of Case 1 (Appendix A.4). Furthermore, it is equally likely that Case 2 or 3 could result from Case 3 after additional kicks.)

Case 3 is very important because it is the secondary creation process that creates a resonance leading to the growth of the electron cloud. If, in timeRemaining, the secondary electron drifts to a position that is the same distance from the beam as the original electron was, then a resonance is created along the present "path."

A Case 3 scenario involving multiple kicks always results in a Case 1, 2, or 3 scenario for the secondary electron. Likewise, successive kicks after a Case 2 scenario will always produce a Case 1, 2, or 3 scenario for the secondary electron. In Case 2, however, a second kick can result in two intermediate situations. The electron is traveling toward the wall when it receives the second kick from the beam, and the kick attracts the electron toward the beam. If the second kick, dp₂, is less than the present momentum of the electron, p₁, then the kick decreases the momentum of the electron. This is Case 2a and leaves the electrons’ momentum after the second kick, p₂, as its original momentum minus the second kick (Appendix A.6). In this case the electron will continue to travel toward the wall after the second kick, and may enter a Case 2 or 3 scenario. On the other hand, if the second kick, dp₂, is greater than or equal to the current momentum of the electron, p₁, then the kick will reverse the electron’s current direction, making it travel toward the beam with momentum, p₂, which is given by the difference between the second kick and the current momentum of the electron (Appendix A.7). This is Case 2b, and, in it, the electron does not collide with a wall or create a secondary electron (Figure 9). In this case, the electron will travel toward the beam and enter a Case 1, 2, or 3 scenario when the next bunch arrives. Ultimately, an electron in a Case 2 scenario will eventually enter a Case 1, 2, or 3 scenario after additional kicks. For all kicks or combinations of kicks, an electron’s motion can always be described by Case 1, 2, or 3.

The computer programs were run for various initial conditions. The calculations from the programs were checked against manual calculations, and were determined to be accurate.

I also thank the United States Department of Energy for giving me the opportunity to participate in the Energy Research Undergraduate Laboratory Fellowship (ERULF) program, the National Science Foundation for its help in funding the program, and Argonne National Laboratory for allowing me to conduct research at such a prestigious institution.

The research described in this paper was performed in the Accelerator Systems Division of the Advanced Photo Source, located at Argonne National Laboratory, and was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. W-31-109-ENG-38.

Appendix A:

Click here to view Appendix A