Dynamic of Electron Cloud

Fermilab Main Injector

Complex problems have simple, easy to understand wrong answers
from Murphy's laws



0. Table of contents

1. Introduction

1.1. Setup

Fig. 1.1.1
Let us consider a cylindrical pipe with radius r and charged particles (protons for definiteness) that moves along this pipe. The setup shown at Fig 1.1.1. Pipe contours represents by blue lines and proton shown as pink cylinder. Proton beam has given linear density λp and rms radius σr. Density and rms radius of a proton beam is a periodic functions of time. The beam consists of protons grouped in bunches.

1.2. Producing of electron clouds

Fig. 1.2.1 Fig. 1.2.2
The electrons in accelerator pipe can be produced in the three ways:

  • By collision lost protons with pipe surface. It shown at Fig 1.2.1.
  • By proton collisions with residual gas in pipe (Fig 1.2.2).
  • By collision electrons with pipe surface (Fig 1.2.3).

  • Electrons that produced by protons called primary, and electrons produced by another electrons called secondary. The frequency of producing primary electrons depends only on linear density of proton bean (roughly its a bunch population divided by bunch length). The frequency of producing the secondary electrons depends on density of all electrons ρ, pipe radius r, and average velocity of electrons.
    Fig. 1.2.3

    1.3. Characteristic quantities

    Table. 1.3.1
    Parameter Symbol Value
    Pipe radius r 12.5 cm
    Bunch population Nb 1013
    Bunch transverse size (rms) σr 1.2 cm
    Bunch length (rms) lb 100 m
    Protons energy ε 10 GeV
    Circ. length L 1200 m
    Harmonic number Nh 6
    Number of primary electrons per unit of time
    per unit of length (at the chamber surface)
    Ps 8.811 sec-1 m-1
    Number of primary electrons per unit of time
    per unit of length (in proton beam)
    Pc 0

    The characteristic parameters for the accelerator is given by Table 1.3.1. From this table we can obtain that protons move with speed of light, the linear density of protons beam is equals to

    λp = Nb / lb = 1011 m-1,
    the protons are present at a location for
    τp = lb / c = 3 10-7 sec,
    the whole period (including time when protons are present and protons are absent)
    T = Lb / cNh = 6 10-7 sec,
    where c is speed of light.
    We take a copper as a material of pipe.

    2. Calculations

    2.1. Physical model

    We assume that:

  • Electrons does not affect the proton's trajectories
  • The dynamic of protons beam can be assumed as quasistatic and interactions as instantaneous. It means that we fix a time calculate this forces at this time and it seems reasonable because the radius of pipe r much less then bunch length lb.
  • The bunch linear density (as a function of time) are given. We use rectangle, sine, and exponential forms of bunch.
  • The count of primary electrons per unit of time is proportional to the linear density of proton beam at this time. The constants of proportionality are given.
  • The energies and angles distribution of primary electrons are given. We use a cosine distribution for angles and Dirac's δ-function distribution for energies.

  • We interested in:

  • The dynamic of electrons. Understanding how it associated with form of bunch and so on.
  • The space distribution of electrons.
  • The value of saturation of electron density and its fluctuations.
  • 2.2 Calculation scheme

    The calculations are realized in numerical way by generating many electrons and solving the differential equations for moving of this electrons.

    2.3. Program & Modules

    For numerical calculations we use a cmee library, version 1.1.

    2.4. Program sources

    You can download the program c source here.

    3. Results

    3.1. Dynamic

    Fig. 3.1.1. Trajectory of electron with rectangular bunch form
    Fig. 3.1.2. Dynamic of electron density with rectangular bunch form
    For first let us consider the accelerator with rectangle bunch form. It is a proton beam with periodic density function (period T). When time t (within this period)

    0 < t < τp,
    the proton linear density equals to λp, and 0, elsewhere.

    The attraction forces from protons is very high, so the electrons, that generated by ionization of residual gas move only in bunch region and does not produce secondary electrons. We consider only primary electrons generated another way (by collision with pipe surface) and secondary electrons.

    The ordinary trajectories of electrons are shown on Fig. 3.1.1.

    Let us consider one secondary or primary electron generated at the wall. For simplicity assume that the initial electron's velocity equals to zero. The electrons start to move in attractive force from protons in periodicity trajectory. But when we turn off the protons (time τp) the electron has a very big velocity (statistically, due to energy conservation law). When this electron collide with wall, it produces very many secondary electrons and process repeats.

    In the 50s there is a space project (just theoretical). The aim of this project - is to penetrate the attraction from the Sun and leave solar system. The proposition of realization is to use a kinetic energy of the Moon (that attract a space vehicle). We have the same result for many vehicles (electrons) and many Moon's (protons).

    The dynamic of electron cloud dencity is represented at Fig 3.1.2. Every proton bunch at figures 3.1.2, 3.1.4, 3.1.6 has a population 1013.

    Due to this features of rectangular form bunches the saturation of electron density is never coming (in most cases).

    Fig. 3.1.3. Trajectory of electron with sine bunch form
    Fig. 3.1.4. Dynamic of electron density with sine bunch form
    We cannot tell the same when bunch have a sine form (red line at Fig. π). The boundaries of bunch are smoothly, so the electron can make a several "periods" along one bunch. The usual trajectory are shown at Fig. 3.1.3. We obtain the saturation of electron cloud density in this case (with the same bunch population and other parameters as in previous case).

    Fig. 3.1.5. Trajectory of electron with Gauss's bunch form
    Fig. 3.1.4. Dynamic of electron density with Gauss's bunch form
    The behaviour with Gauss's bunch form is very similary to behaviour in previous case.

    3.2. Space distribution

    Fig. 3.2.1. Space distribution
    The obtaining electron density does not homogeneous. The distribution of this density are shown at Fig. 3.2.1.

    The function ρe shown has a normalization condition

    ∫ 2πr ρe dr = 1.

    3.3. Fermilab Main Injector

    Table. 3.3.1
    Parameter Symbol Value
    Pipe radius r 8 cm
    Bunch population Nb 1013
    Bunch transverse size (rms) σr 0.0015 cm
    Bunch length (rms) lb 0.47 m
    Protons energy ε 8 GeV
    Number of primary electrons per unit of time
    per unit of length (at the chamber surface)
    Ps 8.813 sec-1 m-1 for Fig 3.3.1, and
    8.812 sec-1 m-1 for Fig 3.3.2
    Number of primary electrons per unit of time
    per unit of length (in proton beam)
    Pc 0
    Magnetic field B (dipole) and,
    yB (quadrupole)
    0 for Fig 3.3.1 (none),
    0.1 T for Fig 3.3.2 (dipole) and,
    0.41 T/m for Fig 3.3.2 (quadrupole)

    The simulation patameters for Fermilab Main Injector is represented in the Table 3.3.1. From this table we can obtain that protons move with speed of light, the linear density of protons beam is equals to

    λp = Nb / lb = 2.13 1013 m-1,
    the protons are present at a location for
    τp = lb / c = 1.57 10-9 sec,
    the whole period (including time when protons are present and protons are absent)
    T = 1.8 10-8 sec,
    where c is speed of light.
    We take a copper as a material of pipe.


    Fig. 3.3.1. Dynamic electron density in Fermilab Main Injector (without magnetic field)
    Input file [Description]

    Fig. 3.3.2. Dynamic electron density in Fermilab Main Injector (with dipol magnetic field B=0.1 T)
    Input file [Description]

    Fig. 3.3.3. Dynamic electron density in Fermilab Main Injector (with dipol magnetic field B=0.01 T)
    Input file [Description]
    The simulations with Fermilab Main Injector parameters gives us the results shown at Fig 3.3.1. (without magnetic field), and Fig 3.3.2. (with dipol magnetic field). The simulations with quadrupole magnetic field gives us the same seult as at Fig 3.3.2.

    The values of proton density is decreasing for convenience. The magnitude of electron density at plots is gived in m-3.

    Bring to notice that the counts of primary electrons are different at Fig 3.3.1 and Fig 3.3.2.

    In strong magneti field electron motion will be like spiral (with electric field) with acceleration in longitudnal direction and drift with velocity v = cE/B in transverse directions. The "radius" of this spiral is inverse proportional to B. If this radius (as Fig 3.3.2) much less then radius of pipe then most electrons keeps from hitting the wall. Otherwise, the density dynamic will be very similar to the case when the magnetic field are absend (the saturation value of electrons can be smaller because the probability of hitting the wall will increase). The case with count of primary electrons 8.812 sec-1 m-1 and dipol field 0.01 T shown at Fig 3.3.3.

    3.4. Discussion

    Fig. 3.4.1. Charge screening
    We obtain results (numerical) in electron cloud density dynamic. The summary is the following.
    We done:

  • The the estimation of electron density value and dynamic
  • We understand the physics of this process
  • We wrote a simply program, that calculates dynamic of electrons

  • The aims and ideas for future

  • Introduce an delayed potentials
  • Introduce a elliptical pipe and bunch forms
  • Introduce a charge screening in the metal of the pipe. The simplest case of this screening for one electron is shown at Fig. 3.3.1
  • Introduce the relativistic effects



  • Published 22 Sep 2005. Authors: Panagiotis Spentzouris, Eric Stern, Ivan Sadovsky