SLAC-PUB-2498
AATF-80/21
April
1980
(T/E)
BEAM EMITTANCE GROWTHCAUSED BY TRANSVERSEDEFLECTING FIELDS IN
A LINEAR ACCELERATOR*
Alexander
W. Chao, Burton Richter,
and Chi-Yuan Yao
Stanford
Linear Accelerator
Center
Stanford University,
Stanford,
California
94305
**
ABSTRACT
The effect
the emittance
accelerator
solved
uniformly
design
of the beam-generated
of an intense
is analyzed
by a perturbation
accelerated
tolerance
bunch of particles
in this
paper.
method for
beam.
transverse
for
The equation
are applied
the recently
fields
in a high-energy
of motion
cases of a coasting
The results
specifications
deflecting
on
linear
is
beam and a
to obtain
proposed
some
SLAC Single
Pass Collider.
Submitted
*
**
Work supported
to Nuclear
by the Department
Instruments
of Energy,
Visiting
scientist
from the University
Hofei,
People's
Republic
of China.
and Methods
Contract
of Science
DE-AC03-76SF00515
and Technology,
-2-
1.
Introduction
Irthis
paper we examine
fields,
generated
through
a conducting
ticles
which
devices
sors
by the passage
pipe,
generated
determining
at their
collision
point
intense
bunch required
ance of these
the walls
which will
intense
verse
charge
deflect
particles
dimensions
are large
wake field
will
of the bunch are deflected
ture.
Thus,
travels
of the pipe
bunch of particles
transverse
linear
reaction
during
colliders
off-axis
and leaves
traveling
travels
energy
rates
of microns
and the mini-
the acceleration
can degrade
in
colliding
to the emittance
down a pipe,
behind
of
of the
the perform-
behind
through
be such that
further
as the bunch travels
phase space occupied
the point
a structure
will
2)
all
particles
down along
interacts
wake field
charge.
the pipe,
the particles
If
an
whose trans-
of the bunch,
away from the axis
by all
it
a transverse
compared to the length
in transverse
increase.
high
devices.
When a point
with
for
succes-
of the order
related
growth
very
in
beam
These linear
dimensions
useful
important
colliding
achieving
is directly
emittance
is
1) as possible
system.
to achieve
dimension
Significant
problem
of linear
for
beam transverse
mum beam transverse
the beam.
rings
center-of-mass
require
bunch of particles
much attention
beam storage
deflecting
of the same bunch of par-
This
to the luminosity
the electron-positron
beam devices
of an intense
fields.
are now receiving
to colliding
of transverse
on the emittance
these
the limits
which
the effect
behind
the
the head
of the structhe total
area
in the bunch
-3-
We examine
appliGtion
the effects
to linear
tions
of motion
tions
by a perturbation
uniformly
for
analytic
are applied
required
for
dimensions
relative
and then solve
Finally,
for
bunches
We calculate
were zero.
x(z,s),
as a function
and s, the distance
of the bunch is very
.
~(2,s)
is
The transverse
with
wake,
the results
an
of this
of an intensity
and as if
the displacement
much less
is uniform
the displacement
of
of z, the longitudinal
from the beginning
of zero transverse
wake field
the bunch at the position
charges
equa-
beam and a
of the bunch (z is positive
The approximation
Thus the transverse
effect,
these
the bunch as relativistically
bunch is a good one in most cases of interest
dimension
strong
the equa-
beam applications.
to the center
head of the bunch),
accelerator.
find
the cases of a coasting
found. 3)
is
particular
We first
For the case of a very
we treat
on the bunch,
position
beam.
with
of Motion
transverse
a point
method for
colliding
In what follows
its
accelerators.
to the SJAC linac
linear
Equations
wake fields
in the bunch,
solution
analysis
2.
electron
particles
accelerated
asymptotic
of these
toward
the
of the
dimensions
for'the
where the transverse
than the size
of the pipe.
across
the bunch.
of the center
of a slice
In
through
z.
force
at z depends on the displacement
z' > z and is given
of all
by
00
Fx(z,s)
=
e2
dz'p(z')W(z'-z)x(.z',s')
/
z
(1)
-4-
where p is the line
normalfzed
to the total
the transverse
axis
S
density
field
produced
to the retarded
charge
time
the average
W of the structure
the case in electron
The equation
d
ds
charge,
for
of a particle
N),
displaced
the point
the displacement
s so that
(/pdz
in the bunch,
by a point
z' - z behind
' = s - z' + z refers
with
in the bunch
number of particles
by x at a distance
assumed that
of the particles
is
e.W.x
is
from the
and
the field.
We have
changes sufficiently
slowly
can be used (certainly
linac).
of motion
for
x(z,s)
y(s)x(z,s)
] + ($-$
can be written
as
= r o idn'p(z')W(z'
-z)x(z',sl)
(2)
.
where y(s)
being
is the energy
the rest
length
mass of the particle;
of betatron
classical
radius
focusing
focusing
from a series
bunch length
retardation
eq.
W, y,
of widely
wave-
s; and ro= e2/mc2 is the
solve
beam in which
by a smooth function
spaced quadrupoles.
is much shorter
first
is the instantaneous
X, and p are known functions,
is provided
(2) can be replaced
coasting
at position
can be ignored,
We will
X(s)
2
of mc , m
s in units
of the particle.
We assume that
betatron
of the beam at position
than
i.e.,
x(z'
,s')
the
than
coming
rather
We also
the betatron
and that
assume that
wavelength
on the right-hand
the
so that
the
side
of
by x(z',s).
eq.
(2) using
a perturbation
the bunch has constant
energy
method for
as a function
a
of s.
-5-
3.
Coasting
%r
pendent
Beam Solution
a coasting
of s.
placement
beam we have y(s)
Consider
of x=x0
the zeroth
order
a bunch injected
and a slope
solution
for
x1= 0.
We expand x(z,s)
into
ko= 2~/h .
, 0
in a series
= X0 both
the linac
In the limit
the bunch motion
xb) (2,s)
where we have defined
= y, and X(s)
with
indea dis-
of no wake field,
is simply
= xocoskos
(3)
of powers of the wake field
co
x(z,s) = c ,(n)(2,s)
n=o
(4)
.
and obtain
the n
th
order
term fromthe
d2 ,W (z,s) + kix+z,s)
= >
ds2
which
is a direct
xcn)(z,s)
=
consequence
fds'G(s,s')>
0
0
with
G(s,s')
the Green's
(n-l)th
order
term by solving
jdz'p(z')W(z'-z)x("-l)(z',s)
0 z
of eq.
(2).
The solution
rdz'p(z')W(z'-z)x("-l)(z',s')
z
(5)
of eq.
(5) is
(6)
function:
G(s,s')
=
$
sinko(s0
s')
(7)
-6-
Inserting
x (0) from eq.
(31, we obtain
x(l)(z,L)
xo(&)($
by iteration
h
=
sin,,)
. [dzlo(nl)W(zl-
L
coskoL + -8k
0
0
co
co
.
sink
z>
- z> I dz2P (z,)W(z,
“1
I dzlP (zl)W(zl
z
(8)
- zl>
. . .
where we have taken
field,
one can ignore
the deviation
1<<Ix
0
all
higher
from its
is sufficient
1 which
at the end of linac
the terms
of a particle
It
by x(l).
lx(l)
values
than
zeroth
to keep only
s=L.
For a weak wake
the first
order
order
trajectory
the first-order
and
is given
term
if
requires
Lro
m
I dzlo(zl)W(zl2yoko 2
The fact
that
of the resonant
xb)
a cos(kos),
the sketch
along
should.
tion
If
1.
and
orders.
pictures
consequence
and
of the bunch looks
of x (1) depends on the position
at the head of the bunch x
the wake field
to higher
of snapshot
The value
to s is a direct
Since x(-l' a s sin(kos)
situation.
a series
in fig.
the bunch,
Ix (1) 1 is proportional
driving
(9)
z> << 1
is not weak,
(1) vanishes
one has to carry
like
z
as it
the calcula-
-7-
In many practical
bunch?;as
i.e.,
completed
koL >> 1.
reduces
situations,
many betatron
In this
case,
is long
oscillations
we find
enough so that
before
after
reaching
some albegra
that
the
s=L,
eq.
(8)
to
,b) (z,L)
where Rn is defined
Rn(z)
the pipe
z
x0(-&y
eikoL
. R,(z)
(10)
by
dz2P (z2)Nz2-zl>.
dzlP (z,)W(z,-d
=
$- ($)n
z
.. I
z
s1
dznp (~~>“(z,y-z~-~)
n-l
(11)
It
is understood
Note that
that
x (n+~,s)
the variables
A closed
rectangular
only
the real
h as the useful
s and z, although
of eq.
property
x(z,s)
form can be obtained
distribution
part
for
=
N/R
1
0
it
is meaningful.
is factorizable
Rn if
we approximate
function,
for
jz[
< R/2
for
Iz]
>= R/2
p by a
i.e.,
(124
w = w,z/I1
Both approximations
to allow
order
tion
a good assessment
terms.
for
are close
In particular
the SLAG linac.
(12b)
enough to reality
in many applications
to be made of the importance
the linear
We find
in
is not.
and W by a linear
P
that
(10)
wake is a quite
of higher
good approxima-
-8-
(13)
and
x(z,L)
=
(14)
xoe
where we have defined
(15)
The validity
condition
becomes In/41 CC 1.
expression
for
eq.
For intermediate
(9) f or the first-order
eq.
In the limit
approximation
InI >> 1, one can find
an asymptotic
(14):
values
of
lnl
, the power series
expression
(14)
is
more accurate.
4.
Accelerated
Beam Solution
We assume that
as a result
the energy
of acceleration
the beam energy
yomc2
We assume that
with
constant,
h(s)
so that
= X0 .
in such a way that
at injection,
the strength
beam energy
of the beam increases
y(s)
linearly
= y,(l+Gs),
and G the acceleration
of the focusing
the instantaneous
force
betatron
with
s
with
gradient.
in the linac
wavelength
scales
remains
-9-
We first
make a change of variable
from s to a new variable
u= l+Gs
2
ax
-+Ldx+
u du
du2
where we have defined
by an iteration
order
solution
at s=O
by setting
the initial
-z)x(z',u)
equation
as in the coasting
is obtained
zero and demanding
This
x(z,u)=xo
(17)
again
be solved
The zeroth
beam case.
N(k)J
= x0 . -q
side
of
(17)
to
and dx(z,u)/du=O
(&)-Jl(+)No(;u)
where J
If
0' J 1' No'
and N are the usual
1
we expand x(z,u)
obtained
the x's
from
the
(n-l)th
on the left-hand
hand side
replaced
a Green's
x(~)(z,u)
order
side
(4),
- Jl(>&m
Bessel
functions.
the n
term x (n-1)
replaced
by x (n-1) .
order
by solving
by x (4
The solution
th
term x Cd can be
eq.
(17) with
and the x on the right-
can again
be obtained
function:
= J"du'6(u,u')
1
the Green's
as in eq.
(18)
O kG
Nl($)Jo($)
with
(2)
or u=l:
x(O) (z,u)
using
will
the right-hand
conditions
of motion
(2) then becomes
dz'p(z')W(z'
ko=2a/Xo.
procedure
Equation
.
x=------
in the equation
function
"
yoG2u'
rdz'p(z')W(z'z
z)x(n-l)(z',u')
(19)
-lO-
-
G(u,u’)
=
In practice,
+
.[No(+
over.most
betatron
oscillation
required
to double
eqs.
and (20)
argument
x(O)
i.e.,
for
(z,s)
=
in the Bessel
cos
pi
G(u,u’)
The factor
,/l+Gs
The terms
recursion
in eq.
(z,s)
1)
f--
(21)
in the series
,b>
where R,(z)
=
[ I
the distance
that
is then
appear
in
We can use the large
to obtain
J&
the usual
solution
than
the
(211
=
sin[>(u-
is
accelerator,
functions
functions
kO
$U-
(20)
u’)]
The acceleration
than unity.
the Bessel
-
of a linear
ko>>G.
are much larger
expressions
u)N~>
is much shorter
the energy,
and the arguments
- Ji+
ut)
of the length
wavelength
adiabatic
(18)
u)Jo($
=Oskos
u')]
02)
adiabatic
damping factor.
are
=
has been defined
(23)
in eq.
(11) and I,(s)
is given
by the
relation
S
In(s)
=
IO(s)
=
sin
coskos
ko(s1
s')
1In-l(s')
(24)
-ll-
In general,
Howevm,
the solution
if
the beam energy
than the beam energy
solved
to eq.
(24)
for
In is rather
complicated.
at the end of acceleration
at injection,
i.e.,
(l+GL)
is much higher
>> 1, eq.
(24)
can be
to yield
I,(L)
where taking
the real
Comparing
the coasting
results
=. $-
part
beam results,
(25)
eq.
beam results,
eqs.
(23) and (25),
(lo),
that
the accelerated
we note
from the coasting
substitution
beam result
to all
with
beam
orders
by
rule
Coasting
Beam
constant
energy
injection
!?n(l+GLgn
is understood.
the accelerated
can be obtained
the simple
e ikoL[&
Accelerated
--+
y,
displacement
x
injection
Beam
energy
y,
+
0
length
of linac
L
(26)
where the last
lation
rule
L+$
phase exp(ikoL).
2
is y mc .
f
result
replace
wake field
apply
energy
beam result
to the betatron
oscil-
at the end of acceleration
reduces
to the coasting
beam
G-to.
For a bunch with
linear
does not
The particle
The accelerated
in the limit
Rn(yf/yo)
a rectangular
approximation,
x0 by (yo/yf)
l/2
charge
eqs.
distribution
(14) and (16) still
x0 and n of eq.
(15) by
and in the
hold
if
we
-12-
Lr NW
n
In fig.
koL=O,
that
5.
=
ko(y;-
2 we have plotted
~r/2, n, and 3~/2
the value
clear
2 that
Misalignment
analysis,
the effect
cision
and it
travels
of low beam intensity.
position
that
a displacement
down the linac
We will
is such
to 150.
It
large.
into
error.
the linac
in a straight
is adiabatic.
dz'p(z')W(z'-z)
The accelerator
position
will
error
be treated
In this
with
line
the acceleration
can be written
as a consection,
of the accelerator
the case with
of motion
the accelerator
is produced
study
is the transverse
s.
of
strength
of the bunch can be very
injected
d2x
-+Ldx+
u du
du2
where d(s)
values
of the bunch is equal
caused by misalignment
We assume the beam is
The equation
The wake field
and the wake field
with
pipe.
the approximation
(27)
the bunch for
we have assumed that
aligned
of beam injection
study
tail
( )
along
2~r).
the distortion
is perfectly
we will
(modulus
'
2
lz
--2 R
Effects
In the previous
sequence
x(z,L)
of n at the very
from fig.
structure
Yf
y,) ai1<
0
perfect
pre-
in the limit
acceleration
under
as
[ x(z',u)
of the pipe
t
as N
- d(s)]
(28)
structure
structures,
at
with
the
C
i
th
eq.
structure
(28)
misaligned
contains
by a distance
an additional
comes from the pipe misalignment.
force
d..
1
Compared with
term on the right-hand
eq.
(172,
side
that
is
-13-
The zero-th
order
solution
is asTurned to be injected
of the bunch strictly
line.
without
(28)
error.
is x ('I
= 0 since
The trajectory
x (0) and is therefore
follows
The first-order
alignments
to eq.
perturbation
the beam
of the head
a perfect
term comes solely
straight
from the mis-
d.:
1
r d.R.
x(l)(z,s)
=
-
;
c
i
5;
=
l
sin[ko(s-si'l
(l+Gs)l/2:l+GsZ)l/2
00
Rl'.Z'
l
I
wq
(29)
where the factor
sin
an angular
and the quantity
kick,
Note that
term in this
of this
hand,
with
bation
we do not
driven
=
C
i
(S'Si)
2
"odiRi
2y;kzG
error,
with
value
is driven
is necessary
eq.
(l+Gs)/(l+Gsi)
.
(l+Gs)l/2(l+Gsi)l/2
frequency
apply.
to carry
(29)
into
As
does oscil-
may acquire
in wake field.
by substituting
(11).
On the other
which
of the system and
it
(1) .
xw
to
the first-order
does not
of x
by
in eq.
the natural
condition
up to the second order
term can be obtained
(z,s)
a large
term x (2)
reason,
an injection
driving
expect
of the response
has been defined
by a force
frequency
For this
calculation
order
R,(z)
and the resonant
the natural
amplitudes.
is characteristic
- y)]
the case for
case is not
the second-order
late
xW
unlike
system
a consequence
[ko(s
large
out the perturThe secondeq.
(19).
1cosPo(s -'ii]*
R2(Z)
(30)
If
pipe
we assume the misalignment
structure
to the next,
errors
di are uncorrelated
from one
-14-
<x(1)2>
=
<x(2)2> =
&
R;(z)
<d2> (-$-rEn3(zJ
*R;(z)
(31)
C
where <d2>lj2
that
all
is the rms value
structures
have the same length
have approximated
linac
the sum over
and we have also
The factor
the fact
that
l/NC
If
acquire
i by an integral
in
expression
the
an additional
(12b),
ratio
of <x (212 > to <x(lj2
we can use eq.
by empirically
beginning
tributions
growth
controlling
of the linac.
factor
.
C
In eq.
over
that
(31),
the length
yf>>
is not
of l/N
> under
t o obtain
these
the injection
in eq.
R1(Z)
of
resonantly.
<x (212)
that
C’
(12a)
and a.'linear
the quantities
wake
R,(z).
x
0
first-and
The
is n2/1728.
can be substantially
offset
The corresponding
of the
y,.
driven
assumptions
due to misalignment
we
<x (112 > is a consequence
distribution
(13)
have been obtained
and we have assumed
by x (1) is shown by the fact
resonantly
field
for
perturbation
we assume a rectangular
The emittance
ai=L/N
made the approximation
the first-order
That xC2) is driven
does not
of the misalignment
and angle
reduced
x'
0
second-order
at the
con-
(23):
Jn(l+GL)
2G
x sink L xA
o
o -<'OSkoL
(32)
x(2)(z,L)
=
-15-
By choosing
proper
the first-order
‘X
eq.
if
For example,
is possible
contribution,
(30),
eq.
to cancel
(29),
either
or the second-order
by a corresponding
the second order
contribution
misalignment
term dominates,
choose
0
0
X’
contribution,
(32).
one might
of x0 and x: it
misalignment
misalignment
from
values
an[(l+G~)/(l+Gs~)l
4G
=
diRi
c
i
!?,n2(1+GL)
(l+Gs
i
)1'2
'
0
(33)
so that
angle
the second-order
cancels
required
contribution
the second-order
contribution
x0 and x: have rms values
<xi>
=
from the injection
YflY
3Nc !?n(y,/;,)
-ii-
given
offset
and
The
from the misalignments.
.
by
2
i34)
<d >
0
With x0 and x: given
the sum of misalignment
by eq.
(33),
the first-order
and injection
term obtained
contributions
by
is
Rn(l+Gsi)
Rn(l+GL)
(35)
The rms value
of this
x (1)
<x(l)2>
=
is given
by
Yf
Rn y
0
l
R;(z)
(36)
-16-
which
is
l/3
Thus this
of that
the injection
misalignment
first-order
6.
no injection
the emittance
conditions
contribution
growth
not only
but also
cancellation
effort.
due to misalignment
cancels
significantly
the secondreduces
the
contribution.
Application
to SLAC Linac
The wake field
and P. Wilson. 4)
is linear
collider
the case with
scheme of minimizing
by controlling
order
for
in z.
for
the SLAC linac
In the regions
The relevant
operation
has been calculated
of interest
parameters
yf
for
the wake field
the proposed
single
W(-z)
pass
arel)
N = 5 x lOlo
YO
to us,
by K. Bane
u
= 2.4
x lo3
= lo5
(50 GeV)
3
L=3XlO
(1.2
Z
= lmm
5 -5
W. = 5.9 x 10 m
GeV)
CT = 70 urn
X
m
NC = 240
X0 = 100 m
Under these
(37)
the asymptotic
conditions,
order
to find
cant
emittance
the proper
tolerance
criterion
The value
growth.
-
1
for
case has been shown in fig.
of x(z,L)
z = -0 z. If
a wake field
obtain
a tolerance
not having
to eq.
a signifi-
(27),
is
37
The bunch shape
2\/5
the maximum magnitude
we require
must be used in
L, and 150 at z= -R/2.
94 at z=-0
this
=-
for
of 0, according
at z=O,
Z
expression
2.
are 1.5x0
on the injection
The corresponding
for
displacement
displacement
values
z= 0 and 6.1x0
at z=-oz
of
of
for
of <ux,
lx01 s 11 pm.
we
-17-
This
tolerance
injectj%n
on the injection
stability
since
by a set of static
inc
term rather
(33).
given
offset
,
is
the bunch center
given
<x:>~'~
by eq.
beam size
size
ox at the end of the linac,
<d">1/2
growth
2 l/2
<d >
this
0
by
offset,
optimizing
beam size
is found
to be less
is given
injection
After
<d2>l12.
the beam with
the second-order
of x0 and x:
at oz behind
at z = -oz
for
by con-
growth,
to be 0.25
2 112
the bunch center.
than the transverse
we demand a misalignment
at
<d >
For
beam
tolerance
of
= 0.11 mm.
The effect
of the accelerator
ing the reduction
Since
Since
x'.
of the required
which
at z=-oz,
by injecting
the resultant
(36),
perturbation
2.2.
chaise
= 0.35
conditions,
and 0.62
on the jitter-
For a particle
and angle
0
rms value
the injection
<x("> 2>w
x
The expected
is
for
be canceled
tolerance
can be minimized
the optimum
(34),
can always
by the second-order
term.
dominates,
by eq.
trolling
effect
determined
contribution
eq.
are cominated
(2) to xrms
(1) is about
xrms
The misalignment
empirically
error
the requirement
magnet is + 1 grad.
the first-order
the ratio
example,
sets
The corresponding
kicker
effects
than
x0,
the injection
magnets.
of the injection
Misalignment
error,
in luminosity
the luminosity
reduction
factor
is
misalignment
arising
inversely
R is approximately
R
=
from the emittance
proportional
given
is determined
by examingrowth.
to the emittance,
the
by
(38)
-18-
where <x(~)~ > is given
nosity
reduction
<d2>l12
order
the rms orbit
= 0.1 mm, the reduction
in luminosity
term is
factor
appreciable
affect
taken
in the beam which
different
account
become significant
there
if
to investigate
The third-
will
frequencies
in the bunch,
energies
Since
and thus
differ-
be a Landau damping effect
is the spread
(Ak
0
in betatron
which
in the betatron
0
A numerical
tracking
program
is being
pre-
effect.
Summary
We have studied
tance
growth
We first
then
20%.
For
of the bunch and thus
spread
different
Ak L>v
this
is about
of the spread
have slightly
frequencies,
2 l/2
.
<d >
distortion,
tail
the lumi-
noticeably.
wave number in the bunch).
7.
at the very
comes from the energy
particles
ent betatron
only
the luminosity
We have not
pared
3, we have plotted
fig.
versus
does not
will
by eq. (36).In
of an intense
it
(4),
accelerated
(lo),
and (11).
a substitution
the result
from the coasting
For practical
ing a rectangular
assumptions,
wake fields
on the emit-
in a linear
accelerator.
particles
in the bunch and
cases of a coasting
The coasting
beam result
For the case of an accelerated
rule,
applications,
charge
for
method for
beam.
have found
Under these
of motion
by a perturbation
and a uniformly
of transverse
bunch of particles
set up the equation
solve
by eqs.
the effect
eq.
(26),
that
allows
beam
is given
beam, we
one to obtain
beam result.
we simplify
distribution
an asymptotic
the calculation
and a linear
expression
by assum-
wake function.
of the bunch shape,
-19-
eqs.
(15),
and (27),
(16),
can be obtained
for
cases with
strong
wake
fields?
We have looked
acceleration
effect
at the effects
sections.
conditions
collider.
are applied
We find
in luminosity
injection
less
applied.
The
can be minimized
After
by con-
the minimization
to beam emittance
is given
by
(36).
These results
tion
growth
of the bunch.
rms perturbation
of the linac
method is again
on beam emittance
the injection
scheme, the expected
eq.
The perturbation
of misalignment
trolling
caused by misalignments
angle
than +O.l
to the recently
that
the emittance
will
be tolerable
is within
growth
provided
proposed
SLAC single
and the associated
the jittering
f 1 urad and the accelerator
mm.
pass
reduc-
in the
misalignment
is
.
Acknowledgments
We would
concerning
this
like
to thank
work.
Dr.
Rae Stiening
for
many useful
discussions
-2o-
REFERENCES
1)
Design
2)
R. Helm and G. Loew, The Stanford
Report
R. B. Neal,
also
3)
for
the SLAC Single
New York,
references
W. K. H. Panofsky
quoted
Pass Collider
Two-Mile
W. A. Benjamin,
Project
Accelerator,
Inc.
(1968),
(1980).
edited
p. 173.
and M. Bander,
K. Bane and P. B. Wilson,
unpublished
See
therein.
Rev. Sci,
Instr.
39,
No. 2, 206
(1968).
4)
by
CN-16 (1980).
"Wake Function
for
the SLAC linac,"
-21-
FIGURE CAPTIONS
Fig.
1.
Sketches
four
Fig.
2.
of bunch shape for
instances
Bunch distortion
a case with
weak wake field
at
of time.
at the end of accelerator
ent values
of total
betatron
wake field
strength
parameter
for
phase koL (modulus
Q is taken
four
differ-
27r).
The
to be 150 at the
bunch tail.
Fig.
3.
Luminosity
reduction
ment tolerance
Collider.
2 l/2
<d >
factor
for
R versus
accelerator
misalign-
the case of the SLAC Single
Pass
k,s=O
4-80
3817Al
Fig. 1
100
50
0
-50
- 100
-0.5
4-80
0
z/P
Fig. 2
0.5
381485
R
I
0
I
I
I
I
0
4-
80
100
<
d* > “’
I
I
I
200
(pm)
3814A44
Fig. 3