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Space charge fields of elliptically symmetrical beams
NUCLEAR
INSTRUMENTS
AND
METHODS
130
(I975)
I9-2I;
~) N O R T H - H O L L A N D
PUBLISHING
CO.
SPACE CHARGE FIELDS OF ELL1PTICALLY SYMMETRICAL BEAMS M U R R A Y R. SHUBALY*
Atomic Energy of Canada Limited, Chalk River, Ontario, Canada Received 26 May 1975 and in revised form 18 August 1975 The space charge fields within a continuous beam of charged particles are derived for a set of elliptically symmetrical current density distributions. The application of these solutions in a numerical simulation program is described.
1. Introduction
that anomalously high fields can be generated if the initial particle distribution is not smooth. The technique used in the present work is to model the intensity distribution as a linear combination of elliptically symmetrical distributions for each of which the electric field can be calculated analytically. A weighting factor for each of the distributions required to model the beam is calculated using a least squares fit of the mean and root mean square particle positions in one quadrant of the beam. This report discusses the derivation of the electric field expressions for three density distributions, and the application of this technique in a simulation program. Comparisons with the multiple beamlet type of calculation are also given.
Prediction of the behavior of intense beams of charged particles is difficult because of the problem of calculating the space-charge fields. Solutions of the space-charge problem for beams with axial symmetry are relatively easy, however this constraint prevents the study of particle motion through quadrupole and bending magnets. Kapchinskij and Vladimirskij ~) developed a set of envelope equations for elliptical beams assuming constant intensity across the beam. Sacherer 2) extended this treatment to beams with nonuniform ellipticaUy symmetric intensity distributions. These treatments consider only the behavior of the beam envelope, and further assume that the emittance is constant, or is known as a function of time (or beam position). This assumption is not generally valid. Accurate analysis of the motion in phase space of representative particles requires a knowledge of the space charge fields within the beam. This analysis determines the increase in emittance as well as the change in beam size and shape. Previous numerical simulations [for example Chasman3)] have modelled the beam as a collection of charge filaments, with the field at each filament calculated as the sum of the fields due to all other charge filaments. The major problem with this technique is the amount of computer time required. For 2000 beamlets, l0 seconds of central processor time are required on a CDC 6600 computer to calculate the fields at all the particles. For a 1 m section of beamline, this technique would require over 30 min for the electric field calculations alone. The time scales as the square of the number of particles, severely limiting the accuracy if reasonable computation times are required. As two field components must be stored for each beam position, arrays have to be generated, stored, and manipulated. Another problem with this technique is
2. Electric fields in elliptical beams Consider a charged particle beam with beam current I, axial particle velocity v, and with the intensity distribution possessing elliptical symmetry. That is, the distribution across the beam is of the form n ( x , y) = n
+
,
where a and b are the major and minor semi-axes of the beam. The x-component of the electric field is given by 2) E~ =
I abx 2CoY 0
n(T)ds (a2+s):~(b2+s) ½'
where X2
y2
T - - - + - a2+s
b2+s '
with a similar expression for Ey. These integrals can readily be solved for density distributions of the form n = n o Tk,
* National Research Council Postdoctoral Fellow.
19
k=0,1,2,
..,
20
M. R. S H U B A L Y TABLE 1
where y2
X2
T o = T~=o=~-~+~-~,
Mean position values in the first q u a d r a n t for three distributions, the uniform, parabolic and quartic. Distribution
by use of the substitution r cosh 0 = ( a Z + s ) ~,
1 H .
r sinh 0 = (b2+s) ÷, which gives
n a2-b
r 2 ~
=
crab
.
< x 2>
< Ixy] >
a2 4
ab 2zr
a~ "-3
2ab 3--'-~
12a
3a 2
7"-'-~
"-~
5ab 4~
-< Ixl >
.
2...~.(x 2 y_~2t zrab ka 2 + /
4a 3~
.
8a 5"-~
2, = _.3._3( x 2
and
n
~ab ka 2 +
y2~2
b2/
ds = 2 r 2 cosh 0 sinh 0 dO. For a uniform density across the beam
1 HI
and Poisson's equation ~E x + ~3Ey = p_p__,
~ - - - -
nab
Ox
and
t3y
eo
where the charge density I Exl
--
x
_ _
_
p = (I/o) n.
_
neo v a (a + b) For a = b, the solutions reduce to those for a circular beam.
For a parabolic distribution
n2
nab \ a 2 + -~ '
and
E~2
21
x
F x2
= neoV a ( a + b ) 2 L~a2 ( 2 a + b )
For a quartic distribution
3
y2 2
n3 = - ~
+ b2/ ,
and I x Ix 4 Ex3 = n_ e_o v a_ ( a_+ b ) 3 ~a 4 ( 8 a z + 9 a b + 3 b 2 )
2 x 2 y2 y4 ] +~ b(3a+b) +-~ b(a+3b)~. The y-components of the field are found by interchanging y and x, and a and b in the above expressions. These solutions satisfy Maxwell's equation for an electrostatic field
8E~ t3y
~]Ey = 0,
t3x
3. Application The complete program using this field calculation technique models a beam as a collection of beamlets each carrying the same current, and initially distributed in space to give the desired current density distribution• Only one quadrant of the beam is modelled because the initial distribution is elliptically symmetric and, in all cases considered, the beam remains reasonably symmetric about the original axes. A fitting routine calculates the beam semi-axes (a, b) and the parameters (Ixl>,, , (yZ), from the distribution in space of the beamlets. Using a leastsquares fit, the routine calculates the relative proportions (D1, D2, D3) of the three elliptical distributions (with the same semi-axes as above) required to give the minimum error between the distribution parameters ([x[>, ([y[>, , , <[xy[> for the beamlet model and for the sum of distributions model, with the requirement that DI+D2+D3
-- 1.
Table 1 gives the mean position values in the first quadrant for the three distributions considered. Values for (]YI>, (y2> are found by replacing a by b in the expressions for (Ix[>, (x2). The fitting routine also
SPACE CHARGE FIELDS
calculates the root mean square error to test the validity of the fit. The calculation of the space charge distribution is repeated for each step along the beamline. The space charge forces acting on the particles in each step are thus adjusted for changes in beam size and current density distribution. The program can distribute the beamlets in space to correspond to uniform (nl), parabolic (n2) , and Gaussian (2n~-n2) current density distributions. Tests were made using these distributions to compare the two methods of field calculation; the point-topoint (calculating the field at each beamlet by summing the contributions of all others), and the analytic technique using the elliptic distributions. The analytic technique required less than 100 ms for the calculation, the point-to-point calculation required 50s. The point-to-point calculation also exhibited "graininess" due to a lack of smoothness in the initial distribution, especially for highly elliptical beams, and near the centre of a hollow beam (e.g. a distribution such as n2). In all cases the calculated electric fields agreed to within a few percent, except near the centre of a hollow beam where the point-to-point calculation is no longer accurate.
21
4. Summary This report has described the derivation of the space charge field distribution in elliptically symmetric continuous beams of charged particles. These solutions lead to a means of rapidly and accurately calculating space charge forces in the numerical simulation of charged particle motion in non-axially symmetric systems. The author acknowledges many discussions with C. R. Hoffmann, J. D. Hepburn, B. G. Chidley and J. H. Ormrod. Their assistance and advice are greatly appreciated.
References 1) I. M. Kapchinskij
and V. V. Vladimirskij, Proc. Intern. Conf. on High energy accelerators and instrumentation, C E R N 0959) p. 274. 2) F. J. Sacherer, IEEE Trans. Nucl. Sci. NS-18, no. 3 (1971) 1105. z) R. Chasman, IEEE Trans. Nucl. Sci. NS-16, no. 3 (1969) 202.
INSTRUMENTS
AND
METHODS
130
(I975)
I9-2I;
~) N O R T H - H O L L A N D
PUBLISHING
CO.
SPACE CHARGE FIELDS OF ELL1PTICALLY SYMMETRICAL BEAMS M U R R A Y R. SHUBALY*
Atomic Energy of Canada Limited, Chalk River, Ontario, Canada Received 26 May 1975 and in revised form 18 August 1975 The space charge fields within a continuous beam of charged particles are derived for a set of elliptically symmetrical current density distributions. The application of these solutions in a numerical simulation program is described.
1. Introduction
that anomalously high fields can be generated if the initial particle distribution is not smooth. The technique used in the present work is to model the intensity distribution as a linear combination of elliptically symmetrical distributions for each of which the electric field can be calculated analytically. A weighting factor for each of the distributions required to model the beam is calculated using a least squares fit of the mean and root mean square particle positions in one quadrant of the beam. This report discusses the derivation of the electric field expressions for three density distributions, and the application of this technique in a simulation program. Comparisons with the multiple beamlet type of calculation are also given.
Prediction of the behavior of intense beams of charged particles is difficult because of the problem of calculating the space-charge fields. Solutions of the space-charge problem for beams with axial symmetry are relatively easy, however this constraint prevents the study of particle motion through quadrupole and bending magnets. Kapchinskij and Vladimirskij ~) developed a set of envelope equations for elliptical beams assuming constant intensity across the beam. Sacherer 2) extended this treatment to beams with nonuniform ellipticaUy symmetric intensity distributions. These treatments consider only the behavior of the beam envelope, and further assume that the emittance is constant, or is known as a function of time (or beam position). This assumption is not generally valid. Accurate analysis of the motion in phase space of representative particles requires a knowledge of the space charge fields within the beam. This analysis determines the increase in emittance as well as the change in beam size and shape. Previous numerical simulations [for example Chasman3)] have modelled the beam as a collection of charge filaments, with the field at each filament calculated as the sum of the fields due to all other charge filaments. The major problem with this technique is the amount of computer time required. For 2000 beamlets, l0 seconds of central processor time are required on a CDC 6600 computer to calculate the fields at all the particles. For a 1 m section of beamline, this technique would require over 30 min for the electric field calculations alone. The time scales as the square of the number of particles, severely limiting the accuracy if reasonable computation times are required. As two field components must be stored for each beam position, arrays have to be generated, stored, and manipulated. Another problem with this technique is
2. Electric fields in elliptical beams Consider a charged particle beam with beam current I, axial particle velocity v, and with the intensity distribution possessing elliptical symmetry. That is, the distribution across the beam is of the form n ( x , y) = n
+
,
where a and b are the major and minor semi-axes of the beam. The x-component of the electric field is given by 2) E~ =
I abx 2CoY 0
n(T)ds (a2+s):~(b2+s) ½'
where X2
y2
T - - - + - a2+s
b2+s '
with a similar expression for Ey. These integrals can readily be solved for density distributions of the form n = n o Tk,
* National Research Council Postdoctoral Fellow.
19
k=0,1,2,
..,
20
M. R. S H U B A L Y TABLE 1
where y2
X2
T o = T~=o=~-~+~-~,
Mean position values in the first q u a d r a n t for three distributions, the uniform, parabolic and quartic. Distribution
by use of the substitution r cosh 0 = ( a Z + s ) ~,
1 H .
r sinh 0 = (b2+s) ÷, which gives
n a2-b
r 2 ~
=
crab
.
< x 2>
< Ixy] >
a2 4
ab 2zr
a~ "-3
2ab 3--'-~
12a
3a 2
7"-'-~
"-~
5ab 4~
-< Ixl >
.
2...~.(x 2 y_~2t zrab ka 2 + /
4a 3~
.
8a 5"-~
2, = _.3._3( x 2
and
n
~ab ka 2 +
y2~2
b2/
ds = 2 r 2 cosh 0 sinh 0 dO. For a uniform density across the beam
1 HI
and Poisson's equation ~E x + ~3Ey = p_p__,
~ - - - -
nab
Ox
and
t3y
eo
where the charge density I Exl
--
x
_ _
_
p = (I/o) n.
_
neo v a (a + b) For a = b, the solutions reduce to those for a circular beam.
For a parabolic distribution
n2
nab \ a 2 + -~ '
and
E~2
21
x
F x2
= neoV a ( a + b ) 2 L~a2 ( 2 a + b )
For a quartic distribution
3
y2 2
n3 = - ~
+ b2/ ,
and I x Ix 4 Ex3 = n_ e_o v a_ ( a_+ b ) 3 ~a 4 ( 8 a z + 9 a b + 3 b 2 )
2 x 2 y2 y4 ] +~ b(3a+b) +-~ b(a+3b)~. The y-components of the field are found by interchanging y and x, and a and b in the above expressions. These solutions satisfy Maxwell's equation for an electrostatic field
8E~ t3y
~]Ey = 0,
t3x
3. Application The complete program using this field calculation technique models a beam as a collection of beamlets each carrying the same current, and initially distributed in space to give the desired current density distribution• Only one quadrant of the beam is modelled because the initial distribution is elliptically symmetric and, in all cases considered, the beam remains reasonably symmetric about the original axes. A fitting routine calculates the beam semi-axes (a, b) and the parameters (Ixl>,
-- 1.
Table 1 gives the mean position values in the first quadrant for the three distributions considered. Values for (]YI>, (y2> are found by replacing a by b in the expressions for (Ix[>, (x2). The fitting routine also
SPACE CHARGE FIELDS
calculates the root mean square error to test the validity of the fit. The calculation of the space charge distribution is repeated for each step along the beamline. The space charge forces acting on the particles in each step are thus adjusted for changes in beam size and current density distribution. The program can distribute the beamlets in space to correspond to uniform (nl), parabolic (n2) , and Gaussian (2n~-n2) current density distributions. Tests were made using these distributions to compare the two methods of field calculation; the point-topoint (calculating the field at each beamlet by summing the contributions of all others), and the analytic technique using the elliptic distributions. The analytic technique required less than 100 ms for the calculation, the point-to-point calculation required 50s. The point-to-point calculation also exhibited "graininess" due to a lack of smoothness in the initial distribution, especially for highly elliptical beams, and near the centre of a hollow beam (e.g. a distribution such as n2). In all cases the calculated electric fields agreed to within a few percent, except near the centre of a hollow beam where the point-to-point calculation is no longer accurate.
21
4. Summary This report has described the derivation of the space charge field distribution in elliptically symmetric continuous beams of charged particles. These solutions lead to a means of rapidly and accurately calculating space charge forces in the numerical simulation of charged particle motion in non-axially symmetric systems. The author acknowledges many discussions with C. R. Hoffmann, J. D. Hepburn, B. G. Chidley and J. H. Ormrod. Their assistance and advice are greatly appreciated.
References 1) I. M. Kapchinskij
and V. V. Vladimirskij, Proc. Intern. Conf. on High energy accelerators and instrumentation, C E R N 0959) p. 274. 2) F. J. Sacherer, IEEE Trans. Nucl. Sci. NS-18, no. 3 (1971) 1105. z) R. Chasman, IEEE Trans. Nucl. Sci. NS-16, no. 3 (1969) 202.