BEAM DYNAMICS IN HIGH ENERGY LINEAR COLLIDERS
Kaoru YOKOYANational Laboratory for High Energy Physics
Oho-machi, Tsukuba-gun, Ibaraki-ken, 305, Japanfrequency frep

the luminosity is given byNzp
4 TC CTx C!j·INTRODUCTION(I)

We assume that initially the bunches havegaussian distribution in three dimensions. If
the two bunches are oppositely charged likeelectron and positron, each bunch tends tofocus by the electromagnetic field caused by
the other bunch. This is called pinch effect.
The factor P in the above formula comes fromit. It is unity at the low current limit andusually greater than unity. One can easily see
thatthe equation of motion during
thecollisioncanbescaled
using
twoparameters,namely'disruptionparameters'.
They are defined as

The progress of particle physics has been
pushed up the relevant energy range and it isabout to reach tera electron volts in the near
future. Two types of accelerators to this end
have been investigated intensively, namely
large
hadronstorage rings withsuperconducting
magnets and electron
positronlinearcolliders withhighaccelerating
gradient. For the latter numerous acceleration
mechanisms have been proposed.
More thanlGeV/m might be obtained by plasma beat wave
accelerator
or laser-grating
accelerator.
Other mechanisms such as inverse free electron
laser, two beam accelerator and wake field
accelerator
are claimed to give
severalhundred MeV/m. It seems, however, much morethan a decade of R&D is required in order togive a practical design of these acceleratorswith the beam energy of lTeV or so.

Dx<y)-_2 N Ye <Ta(2.)?fOX (JJ ( CTx + oy ) ,where re is the classical electron radius, athe r.m.s. bunch length and r the beam energy
in the unit of the rest mass. The pinch effectfactor P is a function of D and D. Fig.lshows its behavior at smallxvalues Yof D's,
which is r1timated by a computer tracking. Itis known
that for very large values of D's
the factor P saturates and then drops, but
this region is not important in lTeV collider
because of the beamstrahlung effect described
below.

If one is satisfied with a beam energy oflTeV or less, a relatively low acceleratinggradient of lOOMeV/m is enough to give thisenergy
withthe length lessthan
thecircumference of the LEP ring. It seems that
this gradient is within the range of theconventional acceleration
mechanism.
i.e.
microwave plus accelerating cavities, although
an intensive R&D is necessary for the microwave generator.
'Lasertron' is now underinvestigation as a possible candidate of thegenerator and will be reported by other people
in this conference. Here we discuss the beam
dynamics in this electron-positronlinearcollider to clarify the various problems to
get a high luminosity assuming the field
gradient of lOOMeV/m is realized.

5p4

3The major problems except the acceleration
itself are:ll positron production.
2l design of the damping ring which cools thepositron land/or electron! beam to match thesmallemittancerequiredbythehighluminosity.
3J beam instability in the linac.
4l design of the final focusing system.
5l beam-beam interaction.
We discuss here 3l and 5l only. For theproblem ll we assume that at least one useful
positron can be created from one electron,
which looks feasible. The damping ring has noserious problem. A storage ring with the beamenergy of 2GeV or so is sufficient for lTeV
linear collider. The problem 41 has not yet
been pursued by the author. It is not an easy
problem but here we onlY pick up the problems
for future investigations.

20.25

0.5

24sFig. 1
The electromagnetic field in a bunch causes
not only the pinch effect but the synchrotron
radiation called beamstrahlung. Although thebunch length is very short , of the order ofmm, energy loss is very large because themagnetic field in a bunch amounts to several
hundred Tesla in~~ typical case. The energy
loss is given by ~L1Ebs

EBEAM-BEAM INTERACTION=2

Ye31'/· i

FU2J·(3)whereFrom the physical point of view, our major
concern
except the beam energyt~theluminosity. If the two bunches containing N
particles
eachwithr.m.s.
transverse
dimension ax and (J"Y collide in a repetition

Q = Dy/Dx = 'Yx/GjandFCR) = {311

c. 3'.2 .s-( R= 1)

I. 3b/Rl R. >> j_)(4)

We assume R>=1 in the following. In thisformula, the pinch effect and the quantum
mechanical correction are ignored. The latteris necessary because the critical energy of
the radiation becomes a considerable fractionof the initial particle energy in a 1TeV
call ider.

BEAM INSTABILITY IN THE LINAC
We have not yet pursued which type ofaccelerating cavity to adopt. Here we make acrude estimation of the influence of the wakefield assuming the impedance of the SLAC typecavity.
A particle in the tail of the bunch isdecelerated by the longitudinal wake field
produced by the head of the bunch and at theexit of the linac the tail has an energy lower
than that of the head. This energy difference
between the head and the tail must be lessthan the energy acceptance of thefinal
focusing systr~· say, ±0.5% or ±1%. If weaccelerate 10
particles with r.m.s. bunch
length of 1mm to lTeV through 1km cavity
structures, the maximum deceleration is about
15GeV or 1.5%. However, the greater part ofthis energy spread can be eliminatedby
accelerating the bunch slightly in front of
the crest of the r.f. wave and the residualenergy spread may be ±0.3%. This value is wellwithintherangeofthechromaticity
correction in the final focusing
system.
Hence, as long as the longitudfnal wake field
is concerned, the number of particles in abunch can be increased a little.

The
beamstrahlung brings about
variousundesirable effects. The center-of-mass energyof the collision inevitably becomes to have a
spread of the order of ~Ehs' It also produces
background events on the experiments
andmoreover it makes difficult to handle the beam
after calli s ion.
Asanexample we givearelatively
reasonable parameters of 1TeV x 1TeV collider.
N=1x1o 11 ,frep=400Hz

ax=1.4~tm.

ay=0.18~tm.

R=8,

Dx=1.02,
L=1.7x1o 32 cm- 2sec- 1

az=1mm
Dy=0.13~Ebs/E=O. 085

average number of photons
average photon energy
critical energy
=3.65 TeV 2

2.6/electron
33GeV-200GeVThe effect of the transverse wake field ismore serious. When the bunch is dislocatedfrom the axis of the accelerating cavity, thetail of the bunch feels a transverse force dueto the head. The source of the deviation fromthe axis may be the injection error or themisalignmentofthecavities
andthequadrupole magnets.

<s=c.m. energy square>~srms =0. 46 Tev 2
are estimated through a computer
Thesetracking. The luminosity includes the pinch
factor P=1.34.

If the bunch is injected at a distance of x 0
from the axis, the oscillation amplitude xf 0 ft~etai~lof the bunch after
acceleration is
given by:x:r
1
o. 23
t.3 '7"3

The energy deposited by beamstrahlung is as
much as 3kJ per collision which is four ordersof magnitude larger than that in SLC. BY
comparing Eqs.<1l,(3l and (4l one finds that,in order to reduce the beamstrahlung without
lowering the luminosity, it is necessary toincrease the bunch length and/or to make thebeam flat <large Rl and/or to increase therepetition rate. However, all of these havesevere limitations. The bunch length should bekept short because it causes an energy spreadin the linac and makes it difficult to design
an achromatic final focusing system. Too flat
beam imposes a tight allowance of the error inthe beam position control. High repetition
rate results in an enormous electricity power
consumption. Therefore, smaller c.m. energy
spread can only be got at the expense of alower
luminosity,
though aconsiderable
improvement from the preliminary parameters
above may be expected by a careful design ofthe linac and the final focusing system.

-~Xc

I Yr/ ¥,; ·---e'11/6(5)~ = o.to ll~ L re N WoO'fHere,r 1 and r f are initial and final beamenergy in the unit of the rest mass, L thetotal length of the accelerating structure,"A fl the betatron wave length, \1/n. the slope ofth~transverse wake function ~ the origin;
i.e.

W0 = [ dWT(zl/dz lz=O
If we make A/}=240m by installing quadrupole
magnets at e ery 30m, then we get n-120 and
xftx 0-3. The required r.m.s. beam size at theexit of the·linac is about 90 ~tm <horizontal>
and30 ~tm <vertical l.
Finalasci llation
amplitude x and y given by Eq.5 must be lessthan these ~alues,fwhich give the tolerance ofthe injection error of 30 ~tm <horizontal) and10 ~tm <vertical l.
This is a very
tight
condition but still looks feasible. The major
cause of the error of injection position maybe
the jittering of the ejection kicker
magnets in the damping ring. Hence, if weextract
the bunch from the ring in thehorizontal plane,· the vertical position error
will be much smaller than the horizontal one.

As for the final focusing system we only
count up the problems to be investigated.
1l To what extent can the focusing system bemade achromatic ?
2l
Is it possible to design the strong
quadrupole magnets near the colliding point ?
3lWhat should be done to reduce
thebackground events on the experiments ?
4l
Can the degraded beam after collision be
used for other purposes such as positron
production.
fixed target experiments
andsynchrotron radiation research ?

The effect of the cavity misalignment can be
estimated according to ref.3l. Our parameters
give the tolerances of the horizontal andvertical misalignment of 200 ~tm and 80 ~tm<r.m.s.l, respectively.
312

One sees that the transverse wake field
imposes a very strict upper bound of thenumber of particles to be accelerated. A
possible method to relax its effects irs been
proposed by the members at Novosivirsk . The
transverse blow-up is essentialy a forced
oscillation of the tail by the head. If thebunch has a large energy difference between
the head and the tail, the growth of theoscillation amplitude of the tail may besuppressed owing to the chromatic differencein the betatron oscillation frequency. The
blow-up is more serious when the beam energy
is lower.
Therefore, the relative energy
difference required decreases graduallY fromthe initial value, +10% in their case, as thebeam energy becomes higher. At the exit of thelinacit
may be less than theenergy
acceptance of the final focusing system.
The discussion in this section are still
immature but it may be a guide to design theaccelerating cavities for our machine.

REFERENCES
1l R.Hollebeek. SLAC-PUB-2535, June 1980.
V.E.Balakin and N.A.Solyak, Proceedings of
12-th Int.Conf. on High Energy Ace., p124,
Fermilab, Aug 1983.
2l M.Bassetti and M.Gygi-Hanney, LEP-note 221Apr 1980.
3l A.W.Chao, B.Richter and C.Yao, SLAC-PUB2498, Apr 1980.
4l V.E.Balakin, A.V.Novokhats ky and V.P.
Smirnov, Proc. of 12-th Int.Conf. on HighEnergy Ace .• p119, Fermilab, Aug 1983.

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