On Possibility of Using Electron Columns for Controlled Space Charge
Compensation

*V.Shiltsev*

*[Abstract:]{.underline}* We propose to use electron columns formed by
ionization electrons in a longitudinal magnetic field for compensation
of space charge effects in high intensity proton synchrotrons. Strong
magnetic field is to assure that transverse distribution of electron
space charge in the column is the same as in proton beam. Electrostatic
electrodes are to control the accumulation and release of the electrons.
Ions are not 1agnetized and drift away without affecting the
compensation.

1.  *Concept*

It is known that compensation of space charge forces in high current
proton beams is possible is the amount of electrons stored in the beam
is equal to \[Budker\]:

![](media/image1.wmf){width="2.375in" height="0.9201388888888888in"}

\(1\)

where *η* is relative fraction of the electron charge wrt to proton
charge, or degree of charge compensation ; and *γ* is relativistic
factor of protons. The conditions for effective compensation are that
electron distribution is the same as proton one (preferably in all three
dimensions) and that the system of electrons and protons is dynamically
stable. Using electron lenses is one possibility to achieving that, but
the method is not very simple and has certain limits \[Burov, Foster,
Shiltsev\]. Partial neutralization by ionization electrons -- those
which are born by ionization of residual vacuum by charged beam -- was
tried before with some success by the stability criteria was not easily
assured \[Dudnikov\].

Modification of the latter method of passive neutralization could be
more attractive if both protons and electrons are immersed in the
longitudinal magnetic field which is a) strong enough to keep electron
from escaping from the transverse position they are born at; b) strong
enough to keep the system of e-p stable; d) weak enough to allow ions
escape and not affect the process of charge compensation.

The easiest possibility is to have toroidal accelerator, a la "modified
betatron" \[Jackson, Meshkov\] where solenoids provide beam focusing all
along the beam orbit. In the existing accelerators where high energy
protons are guided and focused by transverse magnetic fields of dipoles
and quadrupoles, there is a possibility to satisfy condition (1) in
average having many space charge (SC) compensating elements around the
ring -- as schematically presented in Fig.1.

![](media/image2.jpeg){width="5.555555555555555in" height="4.0375in"}

Fig.1: Schematic layout of super-period P=8 proton synchrotron in which
each cell consists of sequence of dipoles (Dip), focusing and defocusing
quadrupoles (F, D), and SC compensation sections (CSs, or electron
columns).

The fraction of the ring circumference occupied by electron columns
should be equal to *R=η/η~0~* where *η~0~* is the degree of local SC
compensation, e.g. if *η~0~=1* -- full charge compensation inside
solenoids - then *R=η.*
![](media/image3.jpeg){width="4.561111111111111in" height="3.30625in"}

Fig.2: Layout of the compensation section (CS).

Schematically, the compensation section may look as shown in Fig.2 and
consist of solenoid magnet, a pair of ring- or cylinder-type electrodes
to control the accumulation of electrons, controlled leak to vary vacuum
pressure at he location of CS and vacuum ports for possible differential
pumping. The system of electrodes may be more complex than just two
rings and, for example provide desired distribution of potential along
the z-axis (Fig.3). Voltages can be made time-dependent to track changes
of proton beam paramters (energy, size, etc). The solenoidal field
distribution can be varied too in order to reduce x-y coupling it
introduces (Fig.4)

![](media/image4.jpeg){width="4.923611111111111in"
height="3.5527777777777776in"}

Fig.3: Possible distribution of electrostatic potential in CS.

![](media/image5.jpeg){width="4.584722222222222in"
height="3.3319444444444444in"}

Fig.4: Possible CS B-field configurations.

2.  *Estimates*

For the 8 GeV ring (Recycler or FNAL Main Injector), one needs only
*R=1/γ^2^* =1/80=1.2% of the ring occupied by electron columns with full
degree of compensation *η~0~=1.* For 1.4 GeV ring (CERN PS) *γ=2.4*,
required space is *R=16%* for *η~0~*=1 or 8% for *η~0~* =2.

The time needed to generate enough electrons to compensate proton space
charge -- so call "neutralization time - is equal to \[see e.g.
Shiltsev, 1999\]:

![](media/image6.wmf){width="3.8020833333333335in"
height="1.6861111111111111in"}

\(2\)

For a purpose of SC compensation in average (no tracking current
provile) τ should be of the order of 100 us (characteristic time scale
of beam, parameter variation due to acceleration) or longer -- that
calls for vacuum pressure of *P=*5e-7 Torr or better. There is also a
possibility to attain neutralization time which will allow to have
variable electron density for the head/tail and center of the bunch
τ=100 ns if *P=*5e-4Torr. (All for H2 dominated vacuum. Injection of
other gases via a controlled leaks can be considered for this purpose).

It is important that Larmor radius of ionization electrons:

![](media/image7.wmf){width="2.3375in" height="0.8020833333333334in"}

\(3\)

is smaller than proton beam radius (so electron distribution is not
smeared off the proton distribution). Characteristic energy of electron
is:

![](media/image8.wmf){width="2.3020833333333335in" height="1.41875in"}

\(4\)

where *I*\~10 eV is ionization energy, and U_sc is about 120V for
uncompensated 4A beams. Larmor radius for 10 eV electrons in B=1kG is
about 0.1 mm that is much less than beamn radii in the accelerators of
interest.

Coupling (in the units of tune-split) introduced by one solenoid is

![](media/image9.wmf){width="3.3006944444444444in"
height="0.8548611111111111in"}

\(5\)

or about 0.03 for 1kG 1m long solenoid in 1 GeV machine -- it is not
negligible, but can be either compensated by skew correctors or -- even
easier -- by a intelligent choice of B-field directions in different CSs
(or by local compensation in one CS -- see Fig.4).

I have applied all known formulae from \[Shiltsev et al, 1999\] for
possible electron distortions by elliptical proton beams and stability
of the e-p system and it seems that B=1 kG should be sufficient for
electron columns.

3.  *Next steps*

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a)  consider e-p dynamics in electron columns in detail

b)  consider methods to control longitudinal escape of electrons

c)  consider necessity and feasibility of local overcompensation *η~0~*
    \>1

d)  install a 100-1000 G solenoid over 1-m piece of MI beam pipe, equip
    it with ring electrodes, electron clud detector, electron clearing
    electrode and two RF antennae and peform studies of the charge state
    of an electron column.

I am thankful to Yuri Alexahin for fruitful discussion on this subject
and explanation of the coupling due solenoid (5).

Some references are taken from the list below (thanks to Frank
Zimmermann for pulling them together for a different publication),
\[Shiltsev, et al 1999\] is PRSTAB paper on TEL

![](media/image10.wmf){width="4.301388888888889in"
height="3.5597222222222222in"}