The 9th ~mposium on Accelerator Science and Technology, Tsukuba, Japan 1993

Beam Tracking of a Compact Storage Ring (2)Hirofumi Tanaka
Central Research Laboratory, Mitsubishi Electric Corp.
8-1-1 Tsukaguchi-honmachi, Amagasaki, 661 Japan
30 June 1993

Abstrac t
A beam tracking with a numerical integration
method has been studied for a design of a compact
storage ring. Suitable numerical integration schemes
for the beam tracking have been studied. Estimationsof calculating speeds and accumulation s of calculating
errors have shown that a Runge- Ku tta-Verner method
and a Adams-Mou lton's method are suitable for thebeam tracking. Calculating results with various integration schemes have also shown that the numericalintegration method can simulate an electron beam accurately when calculating errors are taken care of andtracking turns are restricted.

1

Introdu ction

Difficulties of simulating an electron's motion in acompact storage ring using a superconduc ting bending
magnet are as follows:
1. non-linear terms of equations of motions must betreated, and
2. complicated three dimensional( 3-D) magnetic
fields (B., Bz, B 11 ) must be considered.

calculating error and a calculating step. Greenspan[4],Maeda[5], and J.M. SANZ-SERNA[6] have studied
the schemes. J.M. SANZ-SERN A, for examples, had
solved the condition of Runge-Kutta scheme which issatisfied with symplecticity. But these schemes takemuch CPU time and cannot be used for a beam tracking.
Considering practical uses, integration schemes
which are not symplectic must be used to take careof calculating errors and to restrict tracking turns. In
this paper suitable schemes of numerical integration
for a beam tracking are discussed. This paper also
discusses calculating accuracies of the numerical integration method.

Formula tion

2The study has been done with the tracking code
'PROVIDEN CE'[7] which uses the numerical integration method. In this code, the equations of motion areaccurately transformed to a simple form so as to fit to
the numerical calculation as follows:x"

There are several methods to simulate the non-linear
terms of equations of motion :1. the Canonical integration method [1] ,y"2. the Lie transformati ons method [2] , and- 1y+ 2_(2xpa1

3. the numerical integration method [3] .
The numerical integration method has advantage ofbeing able to treat exact equations of motion, being
able to treat 3-D magnetic fields easily. The longitudinal magnetic fields (B.) cannot be considerd with theother methods.
When non-symplec tic algorithms are employed witha large truncation error, a phase space exhibits damping or exploring due to the noncanonica l character ofthe integration algorithm. It is necessary to use a symplectic numerical integration scheme to simulate anelectron's motion for a long time not to take care of a= Ga (ax'B, + x'y'B11 - (y 12 + a2 )B,.)
(2)

1 ).

where p is the radius of curvature of the reference orbit (constant), 1 denote the differentiatio n with s, andwhere a and cl are defined as follows:aX

1 +p(3)_e_. f'x-,2-+-y'_2_+_(1_+_:__)_2.

mvV

p(4)

These equations are precisely integrated by using adjusted step sizes. The 3-D magnetic fields (B., Bz, B 11 )of a bending magnet are accurately simulated.

-315-The numerical integrations and the estimations oftruncation errors are done with the following methods:
1. afourth-order Runge-Kutta method (R.K.
method)[8) with a one-full step two half-stepmethod,

2. a fifth-order Runge-Kutta.-Fehlberg method(R.K.F. method)[8) with a embeded formula,
3. afifth-order
RungeKutta.-Verner method (R.K.V. method)[9], witha embeded formula,
4. a fourth-order implicit Runge-Kutta method(I.R.K. method)[lO) with a embeded formula,
5. a 12th-order Adams-Moulton's method (A.M.
method)[ll) with a predictor-corrector method,
6. a fifth-order Gear's BDF method (Gear'smethod)[ll) with a predictor-corrector method.
R.K., R.K.F. and R.K.V. methods are explicit RungeKutta methods. These.methods are efficient for nonstiff systems where the derivative evaluations are notexpensive. I.R.K., A.M. and Gear's methods are implicit methods for stiff systems and a system of nonlinear equations must be solved at each step.

3.2

Mitsubishi's lattice

Estimations of calculating errors were also donewith the compact storage ring whose magnetic fields
were not constant. The calculating errors are considered to be larger than errors of the basic model, because beam position's errors of each step are a causeof a change of magnetic field. Lattice of the compact
storage ring is shown in fig.2 which is in the Mitsubishi
Electric Corporation. The energy and the bending radius are 800MeV and 0.593m, respectively. The bending magnet of the ring consists of a pair of bananashaped coils with a field index (about -0.45m- 2 at thecenter of the bending magnet)and an iron core.
Ideal values cannot be determined like the basicmodel. The calculating errors are estimated by the calculating re~ults of R.K. V. and A.M. methods. Thesemethods differ in accumulations of the calculating errors. ·The ideal values are considered to be within orvery near the two calculated results. Tolerances of eachmethod are so selected that calculating errors of thebasic model are within 1 x 10- 10 •Table 2 shows the differences of the beam positons
after 1000 tracking calculated with the two methods(R.K.V.- A.M.) in various initial conditions (xo, Yo)and various multi pole magnetic fields (m =n =d3 B~ ). Tune was selected v.,
1.33, v11 0.43, respectively, because of simulating an ·unstable condition.
When a beam position became larger than x = O.lm
or y = O.lm before 1000 tracking, the tracking wasended (case 9). Table 2 leads that the simulation of
the numerical integration method has a good accuracyof the order of 1 x 10-9 , when calculating errors of thebasic model are within 1 x 10- 10 and tracking turns
are 1000.

=3

Calculatin g results

3.1

Basic modelCalculating errors and calculating speeds are estimated with. changing tolerances of the truncation error!!. Electron's motions in a constant magnetic field
are simulated. The simulating condition is as follows: an electron's energy, a bending radius, an initial position and tracking turns are 0.8GeV, 0.593m,
x00.02m and lOOOturns, respectively. An initial
step size of the numerical integration is lm, and theactual calculating step size is adjusted that the truncation error is within a tolerance. Table 1 shows
beam positions and calculating times after 1000 turns
tracking where the tolerance of the truncation error is
1.0 x 10- 11 m. The calculatings were done with Sun
StationlO (96Mips). An ideal position is x
0.02m.
Each calculating error and its sign are different even
if the tolerance is the same value. The ~esult that theabsolute values of calculating errors are larger than thetolerance is due to accumulations of the truncation errors.
Figure 1 shows calculating speeds as a function ofthe absolute values of calculating errors after 1000
turns tracking. The figure leads that R.K.V. methodand A.M. method are suitable for the numerical integration method and that I.R.K. method cannot use fora beam tracking because the simulating speed is veryslow.

==

4=d.::? ,Reference s[1] R.Ruth,"A Canonical Integration Technique, "IEEE Trans.Nucl.Sci.,NS-30( 4),pp.2669-2671,1983.
[2)
A.Dragt
etal.,
IEEETrans.Nucl.Sci.,NS-32(5),pp.2311-2313,1985.
[3) H. Tanaka et al.,"Beam Tracking of a Small StorageRing,"in Proceedings of the 7th Symposium Accelerator Science and Technology,pp.287-289,0saka,1989.
[4) D.Greenspan,"Conservative numerical methods forf(x),"J.Comput.Phys.,56,1,pp.28-41,1984.
[5) S.Maeda,"Canonical Structure and Symmetries
for Discrete Systems,"Math.Japonica,25,No.4,pp.405420,1980.
[6] J.M.Sanz-Serna ,"Runge-Kutta Schemes for Hamiltonian Systems,"Bit 28,pp.877-883,1988.
[7) H. Tanaka et al.," A beam tracking of a compact storage ring using superconducting bending magnets,", Rev.Sci.Instrum., 63(1 ),pp. 777-780,1992.
[8) P.J.Davis et al.,"Methods of Numerical Integration,"Second edition, Academic Press, 1984.

x=-316-

[9]T.E.Hulletal.,"User'sguideforDVERK,"Department of Computer Science Technical
Report 100 ,University of Toronto,1976.
[10] K.Dekker et al.,"Stability of Runge-Kutta Methods for Stiff Nonlinear Ordinary Equations,"NorthHolland, 1984.
[11] "IMSL MATH user's manual version 2.0" ,pp755770,1991.

Table 1
Beam positions and calculating times after 1000
turns tracking where the tolerance of thetruncation error is 1.0 x 10- 11 m.

R.K.V.

10-5

10- 10ERROR [m]MethodR.K. method
R.K.F. method
R.K.V. method
I.R.K. method
A.M. method
Gear's method

Position[m]0.02- 1.08 x to-n
0.02 + 6.97 X lQ-llo.o2- 7.04 x w-no.o2 + 2.83 x w-s
0.02 + 7.53 x to-s
0.02- 1.29 X 10-6

Figure 1 Calculating speeds as a function of theabsolute values of calculating errors after 1000

Time [s]
130.1
41.7
21.2
87600
17.2
28.0

turns tracking.

.......

Table 2Differences of the beam positons (Di:ff.x,Di:ff.y)after 1000 turns tracking calculated with the twomethods (R.K.V.- A.M.). Initial points are.Bxmaximum and ,Byminimum.

case12345678910xo[mm]

Yo[mm]

m[~)

n[~]

25.0
25.0
25.0
25.0
25.0
50.0
50.0
50.0
50.0
50.0

5.0
5.0
5.0
5.0
5.0
10.0
10.0
10.0
10.0
10.0

0.0
5.0
5.0
10.0
10.0
0.0
5.0
5.0
10.0
10.0

0.0
0.0
100.0
0.0
100.0
0.0
0.0
100.0
0.0
100.0

Di:ff.x[m]-4.97 X 10 -u-1.47 X 10-ll-2.88 X 10-ll
1.85 X 10-ll-1.28 X 10-ll-1.21 X 10- 10-7.04 X 10-ll-8.34 x w-n
1.99 x w- 12
1.11 x 10- 11-317--- --Figure 2 Schematic drawing of the Mitsubishi's
SR ring.

Di:ff.y[m]-1.26 X 10 · l l-2.14 x lo-u
-1.51 x w- 10-3.26 x w-n-8.61 x lo-n-1.62 x w- 11-6.32 X 10-ll
1.33 x 10- 10
1.97 X 10- 12-7.04 X 10-lO

turns
1000
1000
1000
1000
1000
1000
1000
1000
501000