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T’&lS ENVELOPE EQUATIONS WITH SPACE CHARGE'
Frank J. Sacherer
CERN, Geneva, Switzerland
Summary
Envelope
equations
for a continuous
beam with uniform charge density
and elliptical
cross-section
were
first
derived
by Kapchinsky
and Vladimirsky*(K-V).
In
the K-V equations
3re not restricted
to uniformly
fact,
charged beams, but are equally
valid
for any charge distribution
with elliptical
symmetry,
provided
the beam
boundary
and emittnnce
are defined
by rms (root-mean‘This results
because
(i)
the second
square)
values.
moments of any particle
distribution
depend only on the
linear
part of the force
(determined
by least
squares
while
(ii)
this linear
part of the force
in
method),
turn depends only on the second moments of the distribuThis is also true in practice
for three-dimention.
sional
bunched beams with ellipsoidal
symmetry,
and
allows
the formulation
of envelope
equations
that include the effect
of space charge on bunch length
and
energy spread.
The utility
of this rms approach
was first
demonstrated
by L3postolle3
for stationary
distributions.
Subsequently,
Gluckstern4
proved that the rms version
equ3tions
remain valid
for all continuous
of the K-V
In this
report
these rebeams with axial
symmetry.
sults
are extended
to continuous
beams with elliptical
symmetry as well as to bunched beams with
ellipsoidal.
form, and also to one-dimensional
motion.
Yoment
Consider
single-particle
an ensemble
equntions
moment equations,
namely tile equation
for each moment
involves
the higher
moments in an endless
hierarchy.
However,
if the self-force
is derived
from the froespace Poisson
equation,
xF, depends mainly
on the
second moments and very little,
if at all.,
on the lligher
moments.
This will
be demonstrated
in the following
sections.
The remaining
term “Fs is associated
with
cmittance
growth;
we will
avoid considering
it by
assuming
that the rms emittnnce
is
either
constant,
or that its time dependence
2 pr i or-i . Then 2
is given
in terms of x2, z,
two equations
of (4) form
by (5), and the first
set.
They can be combined to give the K-V type
(6)
where
X is
particles
the
D =
that
obey
rms value,
Z = ;/ x2 .
The space-charge
term in tllis
equation
1~3s 3n inIf wo define
the linear
p3rt
teresting
interpretation.
where c(t)
is determined
of the force
Fs(x,t)
as e(t)x,
by minimizing
the dif fercnce
equations
of
(5)
is known
and E(t)
a closed
equation:
for
the
3 fixed
[e(t)x
/
t,
- Fs(x,t))’
where
n(x,t)
n(x,t)
= ,/ f(x,p,t)
dx
dp,
(7)
tllen
(8)
;:zp
(1)
i, = F(x,t)
where F(x,t)
self-force,
trary
particle
includes
both
F = Fe + F,.
distribution
the rms envelope
In other words,
an the linear
part of the forces,
sceuarcs method.
,
the external
force
and the
Averaging
(1) over an arbif(x,p,t),
we obtain
;=;
(2)
p=F=F,,
It is convenient
The assumption
form.
equivalent
to setting
has the form
equation
depends only
determined
by lenst
to put equation
(4) into matrix
of constant
rns emittancc
is
jZs = e(t)Y$.
Then equation
(4)
h = Fo + aFT
where the last equation
follows
because
Fs = 0 by
Newton’s
third
law.
(We neglect
the small magnetic
selfIf Fe(x,t)
is non-linear
forces
due to internal
motion.)
in x, the second equation
of (2) involves
the higher
moments 7=c
x of the distribution.
for linear
exHowever,
ternal
forces,
Fe f -K(t)x,
equations
(2) involve
only
the first
moments x and 5, and therefore
the centre-of
mass motion
depends only on the external
force,
..
x + K(t);
= 0 ,
(3)
and not on the detailed
form of the distribution.
the remainder
of this paper we consider
only linear
ternal
forces.
The second
7
-
moments
of
f(x,p,t)
satisfy
the
7
equations
-y
xp = xp + xp = p
7
where
higher
= 2 2
- K(t)?
+ -XF,
= -2li(t)xp
+ 2 pF, )
the terms3s
and 3,
moments xn and 5.
5 is
are usually
functions
of
This is 3 general
feature
(4)
the
of
the
covariance
matrix
t
xp
(10)
.-I xp 7 1
and F is
1 !
0
F=
In
ex-
=22==2
7
where
(9)
(11)
-K(t)
Equation
M is the
I + F(t)
+ c(t)
0,
(9) is equivalent
to u(t + dt) = Mn(t)MT where
infinitesimal
transfer
matrix
X(t + dt, t) =
dt.
This procedure
is easily
extended
to two
For three dimensions,
dimensions.
the 6 x 6
matrix
includes
cross-correlation
terms such
the 6 x 6 force matrix
F may
5’ , . . . . while
linear
coupling
terms from both space-charge
forces.
The three-dimensional
equivalent
of
1105
and three
correlation
as 5,
include
and external
(9) has
been
gate
effects
forces
tion
zero
tions
incorporated
into program TRANSPORT’ to invcstiboth longitudinal
and transverse
space-charge
in transfer
lines”.
In many cases the external
will
not involve
coup1 ing and the cross-correlaterms between the different
directions
will
be
or close
to zero.
In this
case the envelope
equareduce to the R-V form (h) for each direction.
One-dimensional
-~
_~
envelope
--~
where g = 1 + 2 In (pipe radius/beam
corresponding
envelope
equation
is
(17)
where
N is
the
number
of
particles
X2 = $ [ jk(z,
‘The envelope
equation
= 4ien(x,t)
hl
is
long
in the
only the
and this
is
.
(12)
1
; + K(t);: - AC - $ 5 = 0 )
x
x3
per
as
unit
(13)
area
dimensionless
parameter
u>
2 1 d(x)
dx 7 h(x’)
dx’
0
-Ii
--~
x i =in
1 x2h(x)
dx %
[ iu
I))
uniform,
11(x)
Farabolic
h(x)
specifies
the
=o
for
for
0
C)
d)
~a”SSlZl,
h(x)
I,($
:u~llow,
the values
Envelope
of
\1 arc
= ;
for
In this
given
Table
dz
(18)
for
continuous
beams
--
the
solution
c3
to Poisson’s
equation
+ s) %
is
(23)
’
where
x -< 1
x j 1
T=L+.L.
a2
a similar
(21)
expression
cxz
b2 + s
+ s
for
t
m
1.
”
which
-u
suggests
r
I
cos
e^C
With the
performed
the
change
of
r sin
0 = -A--c--G'
new variables,
giving
the
integration
dr
N is
the
@ can be
dr’
.
(24)
r
can be evaluated
with
the help
.x
n(x,y)
dx dy = ab
number
of
-w
where
Then
(23)
over
n(r”)2nr’
co
/i
0 = ____
cu
The remaining
integrals
of the definition
9 second type of one-dimensional
envelope
equation
arises
in the study of longitudinal
oscillations
of a
bunched beam inside
a conducting
pipe’.
The longitudinal self-field
is determined
by
(22)
variables
0
N=
there-
-w
-XL: =- 4?iea3b2
n(r’)2xr
x
a+b
I
for the range 01 distributions
likely
to be encountered
in practice,
the variation
in hi is negligible
and the rms envelope
motion will
be accurately
described
by Cq. (14) with constant
41, for example Ai 3 l/6.
is
x
dy
(b2 + ,+
co
rllus,
The term z
Y’
m
3A
1
’
equations
,
I
[YiqAl
h’(z)
. ..a
C‘
x
with
fore
in ‘Table
case
x -< 1
x :a 1
-x?,/2
E
= L F2‘,-x2JQ
. 1
2
“i
(10
distribution.
for
- XT)
dz ]
(15)
= +
= ;(I
and
In the absence of cross-correlations
and coupling
terms,
the envelope
equations
have the form (13) where
the space-charge
terms involve
the average
mx and KY.
These averages
will
depend only on the second moments
2 and q and not on the higher
moments Drovided
the
1
a)
bunch
with values
of hz listed
in Table
1. For this case of
a shielded
electric
field,
the envelope
equation
does
depend on the type of distribution.
However,
if the
form of the distribution
varies
only slightly
during
its
evolution,
for example remains
within
the range uniformparabolic-Gaussian,
then the envelope
equation
(17) can
be used with confidence.
in Ay
the
and Acre
h(x) = (l/N)n(x)
For the four distributions
per
-tx>
is
where M is tlw nwbcr
of particles
r; z . Thi.s equation
can be written
where
and the
equations
For a beam in free space that is very
z-direction
and very r;ide in the y-direction,
x-component
of the self-force
is important,
obtained
from the Poisson
equation
ai.
ax
radius),
particle
i
0
n(r’)Znr
per
unit
dr
,
(25)
length.
K
t(z,t)
= -eg
an(z,t)
___
a2
,
(16)
Q(r)
1106
= ab
i
0
n(r’2)2m-’
dr’
(26)
number
is the
becomes
of
within
particles
radius
r,
and
with
(24)
Eq.
the
normalization
u)
10
EL
h(ri)r2
2ea
= x
a+b
2
- Qh-'11
[N
dr’
dr
= 1 .
(37)
(27)
>
/
0
which
is
easily
XE
+
rN%
=-z-.eN2a
x
this
and the
Using
envelope
equations
I.
;
The parameter
X3 depends
only
weakly
on the type
of distribution
as shown
in Table
1. Thus for
practical
distrithe dependence
of the envelope
equations
on the
butions,
type
of distribution
can be neglected.
The same statement also
applies
if cross-correlations
or linear
external coupling
forces
are present;
in this
case
the more
general
matrix
form
(9) of the rms equations
can be used.
integrated,
a + b
expression
Kx(t)x
-
(7-8)
,+,
for
E 2
-.K.
z
e?N
_
-
23
Y)
the
we obtain
Conclusion
1
~
”
=
0
A rather
surprising
and useful
result
has been
found
for
beams
in free
space,
namely
that
the
linear
part
of the self-field
depends
mainly
on the rms size
of the distribution
and only
very
weakly
on its
exact
form.
Using
this
result,
envelope
equations
for
the
rms beam size
have been derived
that
are exact
for
continuous
beams of elliptical
synmwtry,
and in practice
also
valid
for
bunched
beams of ellipsoidal
form.
The
main restriction
in applying
these
equations
is that
the time
dependence
of the rms emittnnce
must be known
2 pricri
.
G+“y
(29)
$- + KY(t)7 - 5
- 1‘2N-i”
Y
=” .
E+$
These
equations
are identical
to the K-V equations
if
7, Ey are replaced
by the physithe ms values
X, E,,
cal boundary
for
a uniform
distribution,
namely
they
are not restricted
to the
x = 2 j;, . . . . However,
K-V distribution
but are valid
for
any distribution
with
the elliptical
symmetry
(1.9).
Envelope
for
equations
The procedure
bunched
beams
in
with
for
bunched
two-dimensions
the ellipsoidal
Possible
uses of the equations
include
the specification
of stationary
or matched
states
in the presence
For example,
tile periodic
solution
c>i
of space
charge.
Eq. (35)
for
alternating-gradient
structures,
including
radio
frequency
cavities,
specifies
the matched
beam
size
(botlr
longitudinal
and transverse)
as a function
The largest
matched
of rms emittances
and intensity.
size
attainable
without
exceeding
aperture
limits
or
bucket
size
determines
a space-charge
limit.
For a
envelope
oscillations
about
beam matched
in this
way,
the periodic
solution
are
suppressed,
although
higher
modes of oscillations
(sextupole,
octupole,
etc.)
nay
Suppression
of the higher
modes will
require
OCC”tZ.
as vet
undetermined,
on the higher
moments
constraints.
of the distribution.
Anotiler
use is the design
of lowenergy
beam transfer
lines.
beams
can
be repeated
symmetry
(30)
The
electric
field
is”
w
&
x
n(T)
= 2neabcx
(a2
+ &(b’
ds
,
+ s) ‘4
+ s)%(cz
(31)
0
where
x2
T=--+
ai
-and with
XC can
x
analogous
be reduced
N is
b*
expressions
to the form
(32)
+L,
+ s
C2
for
+
E
- eNLA3
xc
where
YZ
+ s
x = --
the number
ad
x
of
Y
and
c
=
.
The
term
per
bunclr
and
_-.-g,=; i
h[~T+d~)%[i;
+ sj s ’ (j4)
3 (1+ s)
The integral
in
tic
integrals
of
evaluation
with
easier
and also
velope
equation
1.
F.
Sacherer,
charge,
2.
I.M.
Kapchinsky
and V.V.
Vladimirsky,
Conf.
on High-Energy
Accelerators
tation,
CERN 1959,
p. 274-288.
3.
P.
Lapostolle,
Quelques
proprietcs
essentielles
des
effets
de la charge
d’espace
dans des faisceaux
continus,
CERN internal
report,
CERN/ISR/DI/70-36.
4.
R.
Gluckstern,
Batavia.
5.
K.K.
Brown
6.
F.3.
Sacherer
7.
A.
Sdrenssen,
The effect
of strong
space-charge
forces
at transition,
report,
MPS/int.
MUlFF!67-2.
8.
The
electric
field
for
a uniform1.y
charged
ellipseid
or elliptical
cylinder
is given,
for
example,
by
Foundations
of Potential
Theory
(Dover
Kellog,
Publications,
New York,
1953),
p. 192.
His derivation
is easily
generalized
to include
any
or ellipsoidal
charge
distribution,
as
elliptical
was pointed
out by B. Houssais,
Rennes
University
(private
communication).
(33)
kcx
particles
References
s
(34)
can be expressed
in terms
of ellipthe second
kind,
but direct
numerical
the Gaussian
integration
method
is
quick
and accurate.
The complete
enfor
X is
(35)
where
A3 =-L
3 Jli
u)
w,
h(r2)r4
dr
h(r’)r’
dr
OF
r
dp
(36)
1107
RMS envelope
CERN internal
equations
with
space
repurt,
SI/DL/70-12.
Discussion
and
S.K.
and
at
Howry,
T.R.
1970
SIAC-91
Sherwood,
Proc.
Int.
and Instrumen-
Linac
Conference,
(1970).
Tliis
\:onferencr.
longitudinal
CERN internal