BEAM TRACKING OF A SMALL STORAGE RING
Hirofumi Tanaka, Tetsuya Nakanishi, Soichiro Okuda
Third Group, Advanced Electrotechnology Department, Central Research Laboratory

Mitsubishi Electric Corporation
8-1-1 Tsukaguchi-honmac hi, Amagasaki, 661 Japan

Conventional linear and isomagnetic approximation of beam optics is not sufficient for designing a small storage ring using
superconducting bending magnets. A new tracking program
PROVIDENCE by numerical integration method was developed. In this program the Lorentz equations formulated in thecurved coordinate were transformed to a simple system of equations without any approximation. As a result, the calculating
speed was improved.
This program was applied to the SXLS lattice of BNL's compact storage ring. A banana shape of superconducting coils wasassumed. The ring parameters calculated by a linear optics program were significantly different from the results obtained bythis exact formulation. Beam tracking by this program showedthat the ring with our bending magnet has a sufficient dynamic
aperture.

1

Method2

Abstract

The equations of motion were transformed to a simple form so
as to fit to the numerical calculation. In the PROVIDENCE
these equations are precisely integrated by using adjusted stepsizes and the real magnetic fields of the bending magnet areaccurately simulated. These features of this program will be
described in the following sections.

Formulations

2.1

The Frenet-Seret coordinate system[4] was used. The distance salong the reference orbit is taken as the independent variable ofthe equations of motion. In this paper the reference orbit meansa path of an on-momentum particle through idealized magnets
with no fringing fields. The x and y coordinates measure thehorizontal and vertical transverse deviation of the actual orbit
from the reference orbit.
In the Frenet-Seret coordinate system the transverse equations are obtained from the Lorentz equation[4]:Introduction~ 8 (y'B,-(1+~)By)

S I 1 + 1/ p
IIx +-x - - - p;;zThere are many computer programs for single-particle beam dynamics in cyclic particle accelerators. Most programs [1][2] makeapproximations in order to gain a calculating speed. For example, they use the linear trajectory equations of motion, they
assume the magnetic field in bending magnets as isomagnetic,
and the model of fringing fields of bending magnets is simple.
These approximations are adequate for the large rings wherethe betatron oscillations are negligible compared to the bending
radius and where bending magnets are isomagnetic. But theseapproximations cannot be applied for a small storage ring witharound ten meter circumference.
A new tracking computer code PROVIDENCE for a small
storage ring was developed. The accuracy of this program wasexamined [3] by comparing its calculated tunes and chromaticities with experiment using the LEAR lattice.
This program was applied to an actual small storage ring. The
calculated ring parameters were compared with the parameters
obtained by a conventional program. The dynamic aperture ofthis ring was also calculated by this program.
In this paper the formulation of the PROVIDENCE and thecalculated results will be discussed.

(1)~s ( -x' B, + ( 1 + ~) Bx) . (2)y" +!_X, =!;2

From conservation of energy, the following equation holds
·2

s-v2

(3)

x 12 +y12 +(1+xfp)2'

where pis the radius of curvature,· and ' denote the differentiation with t and s, respectively.
By using Equation (3), and we finally obtain the following
equations:x"

C1 (ay'B, + x'y'Bx- (x 12 + a2 )By) + 2_[2x 12 + a 2[4)y"1c~ (-ay'B,- x'y'Bx + (x 12 + a2 )Bx) + -[2x'y']. (5)paa~awhere a and C 1 are defined as follows:-287-X(6)a _1+pcl -x -)2.
_e_ f'~-f2-+_y_f2_+_(_1_+___

mvV

p(7)

$+-----t:m ~gO.~.O~------~E=~==~t~B=M~l,-~~---~S(m)Figure 2: Field distribution along the reference orbit

Since these equations have simple forms which fit to the numerical integration, the CPU time was significantly reduced. The
calculating speed was as fast as a two-dimensional tracking without this transformation.

2.2

1.5

1.0

0.5

Figure 1: A schematic of the SXLS storage ring

10.0

Algorithm

The equations (4) and (5) are integrated with a fourth-order
Runge-Kutta algorithm. A speed of the numerical integration
would be slow if we used a constant integration step size which
would be small in order to obtain the accurate results. Ourprogram has a function of adjusting the variable step size to theloccil truncation error.

1.5

2.0

X-10.02.3

Magnetic Fields

The magnet fields B, Bx, By in equations (4) and (5) are calculated at every position of the particle. The field is interpolated
by means of the three-dimensional spline functions. These coefficients are in advance calculated from the fileds at the sampled
points computed with a 3D magnetic code.

3Results and Discussions

Our program was applied to a small storage ring of the SXLS
lattice[5]. A schematic of the SXLS storage ring is shown in figure 1. Following the original design, the energy and the bending
radius are 696M eV and 0.6m, respectively.

3.1

Bending Field

Several parameters of the original design are not open. The
most important unknown parameter is the bending field. We
assumed the bending magnet consists of a pair of banana-shaped
coils without an iron core.
The magnetic field of this bending magnet was computed bya 3D magnetic field analysis code. The calculated field on thereference orbit is shown in figure 2.

3.2

Twiss Parameters

Figure 3 shows the horizontal displacement of the closed orbit
from the reference orbit calculated with the PROVIDENCE.

Figure 3: Displacement of the closed orbit from the reference
orbit

When we design good field regions of superconducting bending
magnets, we must take this difference into consideration.
The Twiss parameters of this ring were calculated by thePROVIDENCE and compared with the results by the MAD[l].
Since the MAD can only treat the isomagnetic field, the bending
magnet was devided into subsections.
By the PROVIDENCE {3 functions were obtained from theellipses of the phase space of the small amplitude oscillations.
The betatron tunes were calculated with the fast fourier transformations.
Table 1 shows the horizontal and vertical tunes and the fJ
functions at several points in the lattice. The maximum difference between the MAD and PROVIDENCE is 15 percent.
Although the horizontal tunes agreed each other, the verticaltunes have significant discrepancy.
The differences of the lattice parameters between the resultsof the PROVIDENCE and the MAD may be attributed to theMAD's approximations of ignoring the Bx and B, and identifing
the closed orbit with the reference orbit.
This comparison suggested that the existing simulation codes
like MAD cannot be applied to a small storage ring with superconducting bending magnets.

-288-

40.0

20.0

i

Table 1: Companso n of Twiss Parameter s
diff.
PROV. MAD
s[m]
+0.6%
0.489
0.486
0.0
+3.0%
0.864
0.930 0.837
1.525 + 13.0%(3., 1.400 1.35
2.729 +15.0%
1.931 2.372
+4.0%
2.728
1.994 2.611
-5.9%
6.335
6.729
0.0-1.4%
4.282
0.930 4.342-12.0%
2.502(3y 1.400 2.848-15.0%
1.326
1.931 1.549
-7.2%
1.323
1.994 1.425-0.85%
1.397
1.409
6.8%
0.425
0.398

s 0.0

-"-20.0

-40.0

-10.0

OD

IOD

zao

x lnml
4.02.0;;f.s 0.0;..

3.3

Beam Trackin g-2.0

The dynamic aperture of this ring was calculated by beam tracking. Since the PROVIDE NCE can accurately track particles
with large amplitude oscillations, this program is suitable for
examining dynamic apertures.
The required good field region was obtained by calculating
lOa+ COD by the SYNCH[2]. The good field region at the1center of the bending magnet is -4mm < x < 20mm and-20mm < y < 20mm.
Figure 4 shows the phase spaces at the center of the bending
magnet. The initial point of tracking is x = 20mm and y =20mm which locates on the boundary of the needed good fieldregion. Considering the forms of ellipses of figure 4, we conclude
this ring has a sufficient dynamic aperture.

- 4 .0 -'=zo.:cO:---_~IOD~-~OD-=---:IOD:t-::-----=zo.oy[mmlFigure 4: Phase spaces of the particle with maximum amplitude
at the center of the bending magnet

Refere nces[1] F.Christop h Iselin, CERN-LE P-TH/85- 15, 1985.
[2] A.A. Garren et al., FNAL, FN-420, 1985.
[3] H. Tanaka and T.Nakanishi, Proc. Ann. Meet. of Phys. Soc.
of Japan, 1989.

4Conclu sion

A new tracking program PROVIDE NCE by numerical integration method was developed. The Lorentz equations in theFrenet-Seret coordinate system were transforme d into simple
forms without any approximation. As a result of these simpleformulations the calculating speed was significantly improved.
This program was applied to a small storage ring with superconducting bending magnets whose coils were assumed to havebanana-sha pe. The calculated twiss parameters calculated withthis program were different from the values calculated by a linear optics code. The differece may comes from the considerationof s and x components of the bending field and from the large
discrepancy between the closed orbit and the reference orbit.
The beam tracking by this code showed that this ring may
have a sufficient dynamic aperture.

[4] K.G.Stefen, High Energy Optics, Vol.17, Interscience Publishers, New York, 1965.
[5] M.Q.Barto n et al., BNL 38789, 1986.

1 The required good field region in x-coordinate is not symmetric due tothe difference between the closed orbit and the reference orbit-289-