INS-NUMA-14
A Sextupole Magnet Correction System for TARN
A. Garren* and A. Noda
August, 1979
STUDY GROUP OF NUMATRON AND HIGH-ENERGY HEAVY-ION PHYSICS INSTITUTE FOR NUCLEAR STUDY UNIVERSITY OF TOKYO Midori-Cho 3-2-1, Tanashi-Shi, Tokyo 188, Japan INS-NUMA-14
A Sextupole Magnet Correction System for TARN
A. Garren* and A. Noda
Institute for Nuclear Study, University of Tokyo
, Abstract h ( A study has been made of possible quadrupole and sextupole tuning [<
configurations for the TARN to obtain various working lines in the
tune ^ Vx-Vz diagram, which may be useful in the experiments on injection
and stacking to be done with this facility. These lines will give a
1 large tune spread to stabilize the transverse collective instability.
&-••• I§^ * On leave from Lawrence Berkeley Laboratory, University of California
- 1 - 1. Introduction
TARN is a storage ring built to investigate the accumulation of
heavy ions1^ by a combination of transverse and RF stacking for application
to the NUMATRON heavy ion synchrotron. For both cases, beams are to be
injected at rather low energies (8 - 10 MeV/u), so that the transverse
coherent instability (TCI) may well determine the maximum intensity
attainable. The limiting intensity allowed by this effect is proportional
to the spread of betatron tunes in the beam. This tune spread can be
provided either by sextupoles or octupoles, which give momentum or
amplitude dependent tune shifts respectively. Since RF stacking will result in a large momentum spread, we have chosen to investigate stabilization with sextupoles.
To use sextupole correction, one must choose the momentum-dependent position on a vx-\>z plot or tune diagram of the injection point, the stack and the path traversed during RF acceleration, taking into account the multiturn injection process, and avoiding low order single particle resonances — especially those, with the periodicity of the ring.
In the remainder of this report, working lines and resonances are discussed in Sec. 2, along with formulae relating the chromatic!ty provided by sextupoles and the TCI intensity limit, Sec. 3 discusses options in TARN for sextupole location and strength, and a set of working lines is presented with their respective intensity limits, and Sec. U explains the computations done to obtain such lines.
The main conclusion of this work is that if sextupoles are placed in all locations now available as shown in Fig.l, the intensity stored will
- 2 - be limited by the TCI to about 4 x 109 particles in the case of N5+, or
about 1/4 the intensity that the injection system is designed to provide.
It may be possible later to provide space for additional sextupole. The
larger tune spread attainable after such modifications roughly doubles
the predicted maximum intensity.
2-1. Working Lines
The injection process for TARN is a repetitive process involving multiturn betatron stacking at relative momentum Ap/p = 3.14 % followed by RF deceleration to a stack.3 When complete, the stack will cover
th« momentum range -3.14 % <_ Ap/p _< -0.68 %. The reference orbit, at
the center of the vacuum chamber, quadrupoles, and sextupoles is at
Ap/p = 0. Its tune values Vx, vz is the working point.
In the work discussed in this report we shall describe some examples of working lines designed for injection studies with TARN. These are the locations on a Vx-vz tune diagram of particles covering the momentum range between ±3.14 °i.
These working lines are defined by the working point Vx, Vz, the chromaticities £x, £z, where E, = dv/d(Ap/p) , and the configuration of sextupole correction magnets used to produce them.
Of the working lines studied, the most promising are shown In Fig.2.
Line A is that of an idealized TARN ring having only dipole and quadrupole fields. Line B is obtained by including the intrinsic sextupole fields in the central part and ends of the bending magnets, and represents the uncorrected working line. The remaining, lines include both these intrinsic sextupole fields and those of the correction sextupole magnets
- 3 - and/or pole face windings. In Fig.3, the momentum dependences of vx and
Vz are given for these work lines.
2-2. Resonances
Fig.2 also shows single particle resonance lines given by the relation
£vx + m)z = p ,
where i, m and p are integers. The order of the resonance is
The lowest order resonances, n £ 5, are most serious, especially the
linear (n = 2) and sextupole (n = 3) resonances. Those with harmonic
number p equal to multiples of the ring periodicity N are dangerous
structure resonances. Sum resonance lines (&-m > 0) are at the center
of stop bands in which amplitudes grow infinitely, while difference lines
(£>m < 0) represent interchange of energy between horizontal and vertical
oscillations.
We have tried to construct working lines that do not cross linear or structure resonances, and give as much tune spread as possible in order
to stabilize against the transverse collective instability. The injection points are placed at fractional values such as 2 i (1/4, 1/5, 1/6 . . .) where multlturn injection is possible.
2-3. Evaluation of Chromaticity Due to Sextupole Fields
The transverse equations of motion are
- 4 - where y is x or z, the radial or vertical displacement from the closed
orbit, s is the path length, and the focusing force Ky is given by
1 1 dBz K _ -1- X If V — V V ~ — x p2 » z » Bp dx
The effect of a perturbation k, so that K -*• K + k, is to change the tune
according to1*^
Av = T—
A sextupole has the vertical field dependence
2 Bs(x) = | B"(0)x , Bg'(x) = dBs/dx = B"(O)x .
The closed orbit for a particle with momentum error Ap/p has displacement
Xp = r)*Ap/p and rigidity Bp = (Bp)o(l + Ap/p). Hence the perturbation
strength k corresponding to a momentum error is
f kx = - (K - nK )Ap/p , kz = (K - nK')Ap/p
and dv..
A set of Ns sextupoles of length £g all located at positions with the same values of ru g , and 6, will contribute to the chromaticity an amount
± nf3 h 4IT BP y '
The K term in the integral above is the part of the natural chromaticity arising from gradients, while any intrinsic sextupole fields will contribute through the K' term. The total chromaticity may thus be written
— 5 - ai Si bi Sl
where E,o is the natural chromaticity, for the i sex of
N^ correction sextupqles, and
ai
In order to obtain some particular values o'i £x and E,z, 'cwo independent
families of correction sextupoles are needed. These families can be the
sets just described, where each member of the set has the same value of
all the parameters.' It is also possible to make a correction family by
combinations of sextupole sets with the same strengths S but located at points with different beta-function values. Thus suppose the sets i = 1, 2, . . . m have strength S^ = tj and sets i = m+1, . . ., m+n have strength S^ = t2. Then the corrected chromaticities will be
+ A2t2 ,
Bifci where m+n a± ,. A2 I 2 ± i=m+l m Bl = I bi » B2 " I bi 1I1 i=m+l 1=1 We have used these expanded sets to cover the case of pole-face windings in the dipole magnets, by conceptually dividing each dipole into four parts and treating each part as a lumped sextupole. 2-4. Transverse Coherent Instability (TCI)5^
When the beam as a whole starts to oscillate transversely,
electromagnetic fields are set up that can act back on the beam to
increase the oscillation amplitude. This potential instability can be
avoided by Landau damping, i.e. by providing a spread in the tunes
Vx and Vz. We usf; the treatment of Zotter, who gives the maximum
impedance in either the horizontal or vertical direction for which a beam is stable against TCI in that direction.6' With modifications for
ions of mass number A and charge state q we obtain the following relation
ZQ = 120 TTO, the impedance of space
F is a form factor depending on the momentum distributions,
a value 0.45 is appropriate for our case
N = total number of ions
rc = classical radius for particle of unit atomic mass
2 = -~ £-r , moc = 931.5 MeV eo mo n = nearest integer greater than v
n = 1/Yfc2 - l/yZ > Yt = transition gamma
Ap/p = full width of fractional momentum spread at half
maximum
v = Vx or vz
V1 = £ = dv/d(Ap/p) = chromaticity .
• The largest impedance in TARN is expected to be that of the vacuum chamber, whose value from ref.6 is given by
*7 A") - d-K)A
where R is the machine radius, and a^ and aw are the beam and chamber
half-width or half-height, and 6 is the skin depth. For the energy in
question, the second, wall resistance term can be ignored. Impedance
is also contributed by the presence of RF cavities, discontinuities in
vacuum chamber dimensions and materials etc. However it appears that for
TARN these contribution are relatively small. Combining these formulae,
the stability condition may be written as follows;
N < 1 294 x ipiB AFPVUn-vrt + Vl Ap N< 1.294 10 2(R/v)(1/21/2) p •
For TARN the transition energy yt = l.S>?. Since the injection
energy is y = 1.01 the ring will be operated below transition where
n is negative. Consequently the chromaticity V1 should also be negative,
since otherwise the factor (n-v)r) + v1 will be close to zero for some
value of n.
When the motion is unstable, the e-folding growth time To is
1 To = V" ,
where for a cylindrical chamber the following relation holds;
z v = N3Croq /A / 2R
3. Possible Correction Sextupole Magnet Systems
In order to stabilize the betatron oscillations in both horizontal and vertical directions against the transverse coherent instability,
- 8 - at least two families of sextupoles are needed, one located at 3X > 3Z
positions and the other located at 3Z > 8X positions.
3-1. Lattice structure of TARN and correction by eight SD sextupoles
TARN has eight lattice periods or cells as shown in Fig.1. A
sextupole magnet, can be inserted between the QD and bending (B) magnets.
These sextupole magnets constitute one family of correction elements,
which we designate SD. The strength of these sextupoles will be limited
to a value of 300 kG/m2 and their length is assumed to be 10 cm.
The simplest sextupole configuration possible for TARN consists of
these eight SD sextupoles only, which we call 8 SD, type "L" correction.
Since there is only one degree of freedom, arbitrary chromaticities are
not attainable, but it is possible to make the two chromaticities equal.
From Sec. 2-4, setting £x = E,7, we obtain for this case
= Bp Bj - A1 Nsn
The numerical values are £xo = -5.74, £jzo = -0.25, Aj = 1.33, B: = -3.11,
from which we obtain t. =1.24 m~2, B" = 142 kG/m2 and EL, = EL = -4.2.
When this chromaticity is used with the last formula of Sec. 2-5 we obtain a limiting intensity of 3.7 x 109 for N5+ at 8.55 MeV/u.
3-2. Pole-face Windings in the Bending Magnet
In order to obtain a second family of sextupole correction fields, pole face windings in the bending magnet can be used as shown in Fig.4.
The space available for such coils is 6 mm at the top and bottom of the vacuum chamber, as shown in Fig.4. If the radial current density
- 9 - distribution follows the relation
p(x) = kQ • x(A/m) ,
where x represents the horizontal distance from the center line, the field
produced by this additional current will be
2 B(x) - j p(x)dx *-^f [x£ax -x ] ,
where pQ, g and y~max are the permeability of air, the gap height and the value of x at the position where the winding is terminated. The sextupole strength at the reference orbit, x = 0, will be
B . g
2 As the value kQ is limited to 42000 (A/m ) by current density con- siderations B" will be limited to values less than 15 kG/m2. The correction system using poleface windings and SD sextupole magnets has the merit that it has the same eightfold symmetry as the main lattice of dipoles and quadrupoles. Type "M"' correction uses 8 SD sexuupoles together with the poleface windings and attains highest intensity among the present possible corrections. But such poleface windings would be very difficult and time-consuming to install.
3-3. The SF Sextupole Family
Besides the family of 8 SD sextupoles, another set of lumped sextupoles can be installed next to the QF quadrupoles, in the space between them and the QD quadrupoles. However there is only room in four
— 10- of these spaces for sextupoles, in alternate cells. Consequently use
of such a set of four SD sextupoles will reduce the periodicity from
eight to four for off-momentum particles and increase the number of
structure resonances to be avoided. The length and maximum strength of
these SF sextupoles are assumed to be the same as that of the SD
sextupoles. We have designed two cases of correction by eight SD and
four SF sextupoles, labeled N and P, for which the highest limiting
intensity for 8.55 MeV/u N5+ iu 4.4 x 109 ions — about 20 % larger than
the L-type (8 SD) correction. For this type of correction the width of
the relevant structure resonances, such as v + 4v - 12, should be
calculated.
3-4. Discussion of Working Lines
In Tables 1 and 2 the main features of the working lines ruggested
are given. They are also plotted in Figs.2 and 3. Table 1 gives the
tunes of the injection orbit, reference orbit, and of the stack top and bottom. The stack bottom corresponds to Ap/p = -3.14 %, with the exception of some cases where a smaller momentum deviation was assumed to prevent the stack from crossing a low order structure resonance or linear imperfection resonance. Some lines however involve crossing such imperfection resonances during the RF deceleration process. The vx value of the injection orbit is taken to be vx = 2 ± 1/n where n = 3, 4,
... The analysis of Uef.3 assumed vx = 2-r- and injection efficiency should be satisfactory for other n values, but the lower integers are preferable. Unfortunately the requirements of low n for injection, avoidance of resonances, and provision of a large stack width to combat
— 11 — the TCI are not easily reconciled. Another complication is that the
working lines are not straight — some show considerable distortion —
see Figs.2, 3a-b.
Table 2 lists the sextupole configurations and strengths and
parameters for the working linos affecting the transverse coherent
instability. Chromaticities, momentum width of the stack, and the
maximum number of particles expected to be stable, according to the
formulae of Sec. 2-4. Other relevant parameters for this purpose are
as follows, for Ns ions at 8.55 >3eV/u:
= Y = 1.009, (3 = 0.1345, yt 1-92, A = 14, q = 5, F = 0.45,
R = 5.06 m, eH = 40 irmm-mr, £y = 5 iTmrn-mr, axjjet = 9.5 mm,
a™ =15.4 mm, ax = 25 mm, az = 3.35 mm, axwa^2 ~ 95 mm,
azwall - 22-5 mm, cr - 1.37 x 10 Q -m .
When the stability limit is exceeded, the e-folding time of the
10 TCI is TO = 0.2 sec for 2 x 10 ions, and varies inversely with N.
However the 2 seconds are required for the multi-turn and RE stacking injection process to store this number of ions.
Lines A and B are discussed in Sec. 2-1. Line B corresponds to the uncorrected ring with working point at Vx = vz = 2.25. If this point is shifted so that injection is at v>x = 2.2 or 2.25, it may be used to try injection in the absence of sextupoles. But the very small vertical tune spread should restrict the stable intensity to the low value of
6 x 108 ions.
Line L (8 SD) is discussed in Sec. 3-1, is attractive because it preserves the periodicity of eight and gives close to the maximum intensity possible without major modifications of the ring.
- 12 - Line M', (8 SB, 8 SD) involves adding polaface windings to the dlpole
magnets. The 8-fold symmetry is retained and the expected intensity is
4.5 x 109 ions.
Lines N and P, (8 SD, 4 SF) differ in that the stack is placed below
and above the Vx = 2.5 line respectively. Line P gives the higher
intensity, but its proximity to the third order structure resonance
3vx = 8 may limit its usefulness.
Line R1 (8 SD, 8 SF) requires expanding the ring enough to include
SF sextupoles in every cell and allowing all sextupoles to be about 20 cm
long. Injection is done at vx = 2 - 1/5 and the deceleration carries
the ions through Vv = v, = 2.0 to build the stack in the usual working
square. In this v;ay the predicted intensity rises to 1.1 * 1010 ions,
or half the amount that can be injected.
A similar case like R' but with 8 SD, 8 SB was investigated, but
the necessary sextupole strength from the poleface windings to store
1010 ions appears unattainable.
3-5. Recommendation
To obtain as much flexibility as possible in the present TARN ring, we recommend installation of twelve lumped sextupoles, an SD in every cell and an SF in alternate cells. These magnets should have a 10 cm core length and a maximum strength B" = 300 kG/m2. Working lines B, L, N, P and possibly others to be designed later can then be used.
- 13 — 4. Computational Method and Example
For the purpose of obtaining the chromaticity, a numerical cal-
culation with the recent version of the computer program "SYNCH" waa
executed. In Table 3 the input data used for type P correction are
given. The number of betatron oscillations in the horizontal and vertical
directions (vx and Vz) are adjusted by fitting the field gradients of QF
and Q-Q by the instruction FITQ. The beta and dispersion functions can
be obtained by the instruction CYC and these functions are shown in Fig.5
and the output of the run is given in Table 4. The effect of the
intrinsic sextupole fields in the inner part of the dipole magnet was
modeled by lumped sextupole magnets (SB) distributed in the dipole magnet.
The effects of the end fields of the dipole magnet was modeled by the
thin sextupole magnets (SXL) attached to the both ends of the magnet. The
dipole magnet is represented by the beam instruction B1IL as .BB, The sextupoles are defined by the SXTP instructions. The closed orbits for ions with various momenta are obtained by the operation FXPT. The fractional momentum, —, which is given in the momentum vector PVEC is varied as DP can be changed by the instruction INCR. From the V values for relative momenta —^- of ±0.001, the chromaticities realized by the P correction sextupole magnets (SD and SF) is calculated by differentiating. AR The working line, obtained by varying —*- over the wide range -0.04 <_ Ap/p P £+0.04 in step of 0.005, is shown in Fig.2. In Table 5, typical example of output of the beta and dispersion functions and the closed orbit for the ions with the momentum difference —^ from the central one are given.
- 14- Conclusions
Due to the low velocity of the injected heavy ions, the transverse
collective instability is expected to play a major role in limiting the
number of particles that can be stored in the TARN ring. Therefore it
vill be desirable to provide a system with twelve correction sextupoles
to make it possible to use various working lines with large tune spreads
to suppress this instability. By use of this system intensify limits
of about 25 % of the design capacity of the injection system should be
stable. However we emphasize that there is considerable uncertainty in
these predictions. For example, the details of the beam distribution in
momentum and the nature of the stacking method used might affect the numbers strongly.
Because of these uncertainties, it may be best with TARN to concentrate at first on investigation and control of single particle resonances, beam diagnostics and closed orbit corrections etc. This will be simplified by using a small chromaticity sextupole correction (Type "S" correction) and injecting a relatively small numbtr of ions. Later when more ions are injected some of the configurations giving tune spreads may be employed if the instability appears.
Larger range improvements such as expanding the ring to increase the possible number and length of sextupoles, by means of which the intensity limit might be doubled, should also be kept in mind. Finally, it might be good to also experiment with lighter ions for which the intensity limit would be higher because of their higher velocity.
- 15- Acknowledgments
The authors would like to express their sincere thanks to
Prof. T. Katayama for very useful discussions about the instability, and to Prof. H. Sasaki for his very cogent comments based on broad experience on the most practical ways to bring new machines into successful operation. They are also grateful to Mro. H. Katayama for her careful typing of the manuscript. The numerical calculation was executed with the central computer FACOM M180IIAD at INS.
- 16 - References
1) Y. Hlrao et al., Test Accumulation Ring for NUMATRON Project
— TARN —, INS-NU11A-10.
2) Y. Hirao et al., NUMATRON — High Energy Heavy-Ion Facility —
Part II, INS-NUMA-5.
3) S. Yamada and T. Katayama, Injection and Accumulation Method in
the TARN, INS-NUMA-12.
4) E. D. Courant and H. S. Snyder, Theory of the Alternating Gradient
Synchrotron, Annals of Physics 3^ (1958) 1.
5) L. J. Laslett, V. K. Neil and A. M. Sessler, Transverse Resistive
Instabilities of Intense Coasting Beams in Particle Accelerators,
Rev. Sci. Instr. 36^ (1965) 436.
6) B. Zotter and F. Sacherer, Transverse Instabilities of Relativistic
Particle Beams in Accelerators and Storage Rings, CERN-77-13 (1977)
175.
- 17 - Figure Captions
Fig. 1 Structure of Main Lattice and the positions of the sextupole
magnets to be inserted. B, Qp and Op denote the bending magnet,
radially focusing and defocusing quadrupole magnets, respectively.
Sp and SD represent the two families of the correction sextupole
magnets.
Fig. 2 Single particle resonance lines and the recommended work line.
The bold lines represent the sector resonances for the main
lattice with eight-fold symmetry. The li.ie of vx + 4vg = 12 is
also given by a bold line in order to make caution because it is a
sector resonance for the.off-momentum particle. Line A represents
the idealized case where no intrinsic sextupole components are
included in the main magnets. Line B includes the intrinsic
sextupole components and represents the work line without any
correction sextupole magnets. Line L represents the work line
which provide the same size of chromaticity, (£ - -4.2) both
horizontal and vertical directions with the use of only 8 SD's.
M' represents the work line obtained by the combination of 8 SD's
and poleface winding in the bending magnets (8 SB's). Lines N
and P represent the work lines with the use of 8 SD's and 4 SF's
sacrificing the eight-fold symmetry for off momentum particles.
Line Rf represents the case with 8 SD's and 8 SF's assigning
additional spaces for 8 SF's between the bending magnets and
QF's.
Fig. 3 Dependence of v-values in horizontal and vertical betatron
-18- oscillations on the fractional momentum. Notation of various
lines are the same as in Fig.2.
Fig. 4 Schematic illustration of the poleface winding in the bending
magnet. The current density in the poleface winding is given by
p(x), where x _'.s the horizontal distance from the center line.
Fig. 5 Beta and dispersion functions in a unit cell of main lattice.
The solid, dashed and dash-dotted lines represent beta-functions
in horizontal and vertical directions and dispersion function,
respectively.
Table Captions
Table 1 Various work lines and their v-values.
Table 2 List of work lines combination with the size of chromaticity,
momentum spread in the stack, the strength of sextupole magnet
and the maximum intensity by the point of view of TCI.
Table 3 Input data for the correction P.
Table 4 The output of beta and dispersion functions in both horizontal
and vertical directions and their derivatives obtained by the
instruction CYC.
Table 5 Example of output of twiss parameters and closed orbit for off-
momentum particle (this case is for -2- = 0.03).
- 19- Injection
CONTROL ROOM
Pig. 1 60 in -21 - -z. • 2.8 2.7 • 2.6 • 2.5 - V\ 2.4 - \^\ 2.3 2.2 2.1 2.0 < < > -0.05 0.05 -0.05 p Pig. 3 (a) 2.5 2.5 \ 2.4 " \ 2.4 2.3 2.3 2.2 2.2 - ' \ 2.1 2.1 \ 2.0 2.0 \ 1.9 1.9 V 1.8 1.8 \ 1.7 1.7 \ 1 I \ 1 -0.05 0.05^- - 0.05 0.05- Pig. 3 Cb) \ -22 £ o tn -23- LO -24- Table 1 Injection Working Stack Top Stack Bottom Point Point Line^\ 3.14 % ; o % -0.68 % Variable A 2.07 2.20 2.23 2.23 2.25 2.26 I'll (-3.14 %) B 2.03 2.20 2.24 2 39( 3 14 %) 2.26 2.25 .2.25 l:li- - L 2.18 2.30 2.33 7 LL 2.12 2.25 2.28 2 39 M' 2.15 2.32 2.36 2.11 2.28 2.32 |;JJ (-2.98 %) N 2.15 2.34 2.38 2.09 2.26 2.30 ££ (-2.68 Z> P 2.25 2.47 (-1 14 7> ^(-3.14%). 2.09 2.30 2.34 *-'^ '"> R' 1.82 2.00 7 14 2.50 , ,, . ( 3 64 1.62 -2.00 £JJ (-1.14 %) 2.49 " - ^J 2.32 2.33 S 2.25 2"37 f-3 14 7) 2.13 . 2.19 2.20 (Upper and lower figures represent the values vx and vz, respectively.) 25 — Table 2 Line Configuration Chromaticity Ap/p ^niax (kG/m) A No Correction -4.35 / -1.07 2.5 % 1.23 x 109 No Intrinsic Sextupole B No Correction -5.74 / -0.246 2.5 % 5.91 x 108 With Intrinsic Saxtupole L 8 SD's -4.19 / -4.28 2.5 % 14.46 3.72 x 109 M' 8 SD's -5.81 / -5.75 2.3 % 25.472 4.47 x 109 & 8 SB's - 2.158 N 8 SD's -5.83 / -5.53 2.0 % 22.849 3.75 x 109 4 SF's -15.272 P it -7.13 / -6.56 2.0 % 29.17 4.39 x 109 -25.01 R' 8 SD's -10.6 / -13.2 2.5 % 55.155 1.09 x 1010 & 8 SF's -31.206 S 8 SD's -1.49 / -1.45 2.5 % - 3.505 1.53 x io9 & 4 SF's 30.000 -26 - Table. 3 PACit TARN' ^ NUX»2.V7O» NUi'2.300 • WITH SEXTUPOLE CORRECTIONS. c SOLUTION 2 TYPE-1 C8 SD AND 4 SF) C BRHO =5/9/79 11.451711 RHD =E 1.34403166 flO >C 8.5204U59 DP -0.001 GF 23. GD B -42. BSD 29.1743405 6SF -25.013678 C LS DRF .2458 LF DRF .1642 L DRF .6227 C 6 -IAG .13195 0. BRHO BO SD 5XTP 0. GSD BRHD SF 5XTP 0. GSF BRHU SS 5XTP 0. 0.1064ie80DHHD SXL 5XTP 0. -2.3720027BRHD C .BSB 3ML B S3 S .3B iJML SXL 4C .liSb ) SXL .C 3ML filDri (jDH LS SD LS .33 LF BFH SFH LS SF L L LS (uDH (ODH LS SD LS .33 LF '3FH OFH LS L • L LS c SUB WFDDH •1AG 0.1315 GF BKHO OOH MAS 0.1315 GO BRHO C •(MM • C END c FITS 00 C GF GD 1 1 .6175 .3750 c c c CYC 4 .C SR SUB E PVEC DP FXPT i 4 E .C • STEP • 1NCR 1 DP 0.001 END C CALL 3 SR REPL 1 1)P 0. REPL 1 STEP 0.O05 CALL y SR REPL l STEP -0.005 c CALL 9 Sl< STOP G> Uf > c C p c O C c o c OG o io o o c C o c O O c o o oa o o a o o C o a O C c o c C3 C c o o o c c ,;o c o a c to o o c a o a oo a c oo o c c o o o o a o o oc a !o o o o o 'a o oo o o a o o o o c c o o o a o £> C c ;o a a o a o o >• |m m >• 47 3 3.6 5 TT 70 3 1.1 61 0 1.0 7 52 2 1.1 05 0 2.5 6 97 0 4.5 4 75 0 5.5 3 11 5 5.8 2 43 1 5.5 3 2f 0 TB 8 */> o o o o oo o c o o o o o o o o o o O- CO CO CO ao FM _ > o OCT u H O o an 80 9 4.a s 36 4 3.8 1 50 3 1.9 7 13 5 5-2 9 37 2 3.8 3 33 6 2-4 3 •M 4" 1 0. 0 73 O D O o o D O o o o o D O o o o D O T O n o O CO 83 2 0.1 5 82 4 O.l i 46 3 0.1 2 54 6 0-0 9 50 4 0.1 1 0* 0 M !*- n 2* 0 H O s; 1. 5 i ii x r x n a in < n ft rfl n u. 1 >X L 1. 8 U.IJ- U. I I3 ° n o -* rg o a« •>. m "v *! in D t— O rH rylp >J CM 1 0 5 1 rH|* •I *H! y cjjr fM.C g rvj r I 1* 1 1 -28- Table- 4(continued) 30 LS 4.4332 0.38260 1.658233 -1..37037 0.,952017 0.31598 0.30262 4.533504 1..91359 0.0 0.0 31 SO 4.4832 0.33260 1.658238 -1,,37037 0.952017 0.31598 0,.30262 4.533504 1,.91359 0. 0.0 32 LS 4.7290 0.40211 2.436766 -1,.79696 1.029686 0.31598 0,.31223 3.654911 1,.66083 0.,0 0.0 4.7290 0.40211 2,436766 -1,,79696 1,,029686 0.31598 0,.31223 3.654911 1..60083 0.,0 0.0 34 •i 4.3609 0.4099a 2. 914650 -1,.31311 1..072827 0.33739 0,.31834 3.234521 1,.52515 0.,0 0.0 35 S3 4.8609 0,.40998 2.,914650 -1,.81311 1.072827 0.33739 0 .31834 3.234521 1.52515 0.,c 0.0 36 4.9929 0,,41066 3.,387586 -1,.75959 1,.113580 0.35554 0,.32526 2.849938 1,.30946 0.,0 0.0 37 3 5.1248 0,,4^247 3.,837404 -1..63844 1.166504 0.37027 0.33313 2.J01163 1,.25378 0.,0 0.0 — 3d 5.1248 0.,4?2'i7 3,,637404 -1..63U44 1 ,166504 0.37027 0.33313 2.501163 1.2537b 0.,0 0.0 39 i~ 5.2568 0, 4,.246812 -1 .45433 1 .216137 0.38143 0.34211 2.168194 1.11809 0,.0 0.0 40 S 5.3837 0,.43241 4,.600079 -1.21434 1.267003 0.38392 0.35239 1.911033 0 .93241 0..0 0.0 41 5.3887 0, .60007? -1 .21434 1 .267003 0.38692 0.35239 1.911033 0.96241 0,.0 0.0 42 d 5.5207 0,:«3o'iH 4,.083629 -0.92767 1 .318610 0.39267 0.36416 1.669679 0 .S'-672 0,.0 0.0 43 5.6526 0,,44104 5,.036565 -0.60536 1.370462 0.39263 0.37760 1.464133 0 .71104 0,.0 0.0 44 S3 5.6526 0,,4410* 5,.086565 -0.60536 1.370462 0.39263 0.37760 1.464133 0.71104 0,.0 0.0 c 45 3 5.7346 0..44512 .201069 -0 .25978 1.422059 0.38881 0.39288 1.294393 0.57535 0.0 o.o 46 SXL 5. 0..44512 5,.201089 -0.25978 1.422059 0.381)81 0.39288 1.294393 0.57535 0,.0 0.0 47 LF 5.9488 0.45010 5,.291935 -0.29348 1.485902 0.38681 0.41453 1.133172 0.40651 0.0 0.0 46 ;JFH 6.0803 0,.45407 5.201461 0.97406 1.512764 0.01862 0 .43360 1.079003 0.0093D 0.0 0.0 49 3FH fc.2118 0 .45624 4.790570 2.11679 1.490772 -0.35218 0 .45272 1.127873 -0.33550 0.0 0.0 50 LS 6.,4576 0 .4fo739 3,.819079 1.83557 1.404205 -0.35218 0.48430 1.378911 -0.63581 0 .0 0.0 51 L 7.,0303 0 .50376 1.976678 1.12316 1 .184901 -0.35218 0 .53800 2.565638 -1 .26990 0.0 0.0 52 7.,7030 0 .57597 1.021523 0.41074 0.9655.'6 -0.35218 0 .56720 4.542132 -1 .90411 0.0 0.0 53 LS 7,,9-r88 0.61750 0.883727 0.12952 0 .879029 -0.35218 0 .57500 5.539720 -2 .15443 0 .0 0.0 P35 S PSIX BETAX ALPHAX XE'l OXEB PSIY BETAY ALHHAY YESl DYES 5.06036 THETA- 6.2831853? MX* 2.47000 2.30000 T(;AH*< 2.03713. 0.0 MAXIMA BETAXt 20) 5.29194 XEG 48)' 1-51276 3ETAY( 1)» 5.82705 YE8C 53)» 0.0 MINI !-'A BETAXC 53) 0.81:873 23)= 0•85482 BETAY( 48)» 1.07900 YESC 53)- 0.0 CUNTRI3JTID:;S TO CHR1IMAT1C1TY (JNU/COP/P)) FRUM MAGNETS EXPLICITLY IN CHRX * -4.2341, CHRY -7. 4623 Table- 5 ' CALCULATION CF THE EGUILIBRlUM ORBIT AND BETATKUN FUNCTIONS OF INITIAL REFERENCE RAY DEFINED X - "5.0 DX 0.0 Y 0.0 DY 0.0 OS 0.0 DP/P • 0.03000000 1.00000000 IiTEHATION » 1 XO* 0.028661727 3 DXO* -0.01206534 YO- 0.0 DYO- 0.0 iTEH 0.02633099 0XC1* -O.Ollb3874 0.0 OYU« 0.0 _ ITERATION »_ 3 XU* 0.02632631 DXtjJ 0.0 OYD- 0.0 ITEf 7X7 MATSIX FOR -0.82049347 -0.46525295 0.0 0.0 0.0 1.61229163 -0.00232434 0.36693762 -1.01071029 0 .0 0.0 0.0 -1.23407492 0.00287803 0.0 0.0 -0.63752855 -1.16915317 0.0 0.0 0.0 : o.0 0.0 0.l?<.69703 -1.33620959 0.0 0.0 0.0 -0.42093872 -2.20371544 0.0 0.0 1.00000000 -2.61681158 • 0.00523323 0.0 0.0 0.0 0.0 l.oooooooo 0.0 ~o7B~ 0.0 0.0 o.o o.o "TTuooboooo 'ETSENVALUES OF THE 4X4 SUbMATRIX OX.. Lf'Dl ( -0.91560188 0.40203606 1.00000000, MUC1) 2.72779859 RAD, 9(1) 0.73657052 -0.91550183 -O.« C(2) * 1.00000000, MU<2) -2.72T79859 RAD, OY.. L'-<03 * ( -0.98686907 0.16152225 C(3) l.oooooooo, MU(3> 2.97935969 RAD, 0.89671929 • L«D3 = t -0.98636907 -0.16152225 >, C(4> » 1.00000000, MLK4) -2.97935969 RAD, (SC4) 0.10328O7l_ X DX Y DY DS DP/P EC US61T 0.02832636 -0.01184331 0.0 0.0 0.0 0,.03000000 1.00000000 ET ORBIT 0.99603378 -a.43193300 0.0 0.0 0.0 1,.00000000 0.0 Table- 5(continued) /ECTG!;s i 3 IN PuL,\l< CuiJROl NATES P05L XI DX1 Yl DY1 X3 IDX3 Y3 DY3" ' 0 1.075685 -0.oooooo 0.955293 -l.338528 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.690417 0.000000 0.885660 -0.433088 — —,— BEfAT'SiJN FUNCTIONS UH POS s 'iIX IV iiX BY M i AY EX EXP EY EXP XCO UXCd YCl.1 DYCb CM) CM. TM) CM) (M > (MM) (MR) - (MM) cI-.H) 0 0.0 0.,0 0..0 1 .16 7.,24 -0,.24 -2. 16 0.,99603 -0. 43198 0.0 0.0 28. 3264 -11.,8433 0.0 0.,0 1 QDH 0.13 0.,02 0.,00 1 .30 7. -0..83 0.59 0.96288 -0. 07537 0.0 0.0 27.4597 -1.,3919 0.0 0,0 2 SDH 0.2t 0.03 0.01 1 .•.I 6.94 -1,.60 3.22 0.97608 0.27756 0.0 0.0 27.9573 3.,9903 0.0 0.,0 3 LS c.51 0.05 0.01 2.53 5.,45 -2,.14 2.82 1.04430 0.27756 0.0 0.0 30.1671 8.,9903 0.0 0.,0 4 SO 0.51 0.,05 0..01 2.53 5.,45 -1.95 2.41 1.,04430 0.20076 0.0 0.,0 30.,1671 7.,0649 u.0 0.,0 5 LS 0.75 0,.06 0,,02 3.61 4.,34 -2 .42 2.11 ,09365 0.20076 0.0 0.,0 32.,1003 7.,5649 0.0 0.,0 6 SXL 0.75 0.,06 0..02 3.61 4.,34 -2 .45 2.13 \\.09365 0.20112 0.0 0.,0 32.,1003 7.,9685 0.0 0,.0 7 B 0.,89 0,,0f 0..03 4• 2f 3,.78 -2 .42 1.96 1,.12398 0.22816 0.0 0.,0 33..1929 S.,5806 0.,0 0.,.0 8 SS 0.,39 0.,07 c..03 4.27 3.,7a -2 .42 1.96 1.,12398 0.22783 0.0 0.,0 33.,1929 8.,5756 0.,0 0.,0 9 a 1.,02 0.,07 0..03 4.92 3.,27 -2.31 1. p-8 1,.15700 o.24619 0.0 Ol,0 34.,3bO6 9.,109J 0.0 c,.0 10 6* 1,,16 0..08 G.04 5.53~ 2,.81 -2 .11 1.61 1,.19246 0.26232 0.,0 0.,0 35,.5934 9,.5615 0.,0 0,.0 11 S3 1..16 0..08 0,.04 5.53 2,.81 -2 .11 1.61 1,.19246 o.26194 0.,0 0..0 35,.5934 9,.5558 0.,0 0,.0 " 12" B 1.,30 0,,08 0,.05 6.08 2..39 .84 1.44 1,.23001 0.27569 0.,0 o.-a 36.,8793 •9..9206 0.,0 0.,0 13 3 1.,43 0,.09 0,.06 6.54 2,.02 -1.50 1.26 1,,26936 0.28693 0,,0 o.o 38,,2074 10,.1953 0.,0 0.,0 14 Sb 1..43 0,.09 0.06 b .54 2,.02 -1.49 1.26 1,,26936 0.23650 0.,0 0..0 38,.2074 10,.1887 0.,0 0.,0 15 1.57 0,.09 0.07 6.91 1.70 -1 .10 1.09 1,,31010 0.29514 0.,0 0..0 39,,5649 10..3709 0.0 0 • 16 Q 1.70 0,.09 0.08 7.16 1.42 -0 .66 0.91 1 .35193 0.30109 0.,0 0,.0 40,.9402 10 .4590 0.,0 0,.0 17 SO 1..70 0,.09 0.08 7.16 1.42 -0 .66 0.91 1 ,35193 0.30060 0,,0 0,.0 40,.9402 10 .4514 0.,0 0,.0 18 5 1,.fi4 0,.09 0.10 7 1.20 -0 .20 0.74 1 .39441 0.30332 0,.0 0,.0 42,.3198 10 .4446 0,,0 0,.0 19 SXL 1..84 u,.09 0.10 7.29 1,.A) -0Ut 0. 15 1,,39441 o.31551 0,.0 0,.0 42,.3198 10,.6246 u.,0 0,.b LF .00 0 .10 0.12 .38 0.99 -0 .28 . o.53 1 .44621 0,,31551 0,,0 0,.0 44 .0644 10,.6>246 0,.0 0 .0 21 •3FH .13 0 .10 0.15 7.23 0.90 1.44 0.15 1 .46559 -o.,02281 0,.0 0 .0 44 .7634 -0 .0213 0,.0 0 .0 22 9FH 2,.27 0,.10 0.17 6.64 0.91 2.99 -0. 21 1 .44025 -0.,36042 0,.0 0 .0 44.0588 -10 0,.0 0.0 23 LS 2, 0 .11 0 5 1.OR .6? -0. 49 1 .35166 -0.,36042 0,.0 0.0 41.4370 -10 .6666 0 .0 0.0 • 2.4 SF 2.51 0.11 0.il 5.26 1.U8 2.16 -0.40 1.35166 -0,,24346 0,.0 0.0 41.4370 -8 .84bO 0.0 0.0 25 L 3 .13 0.14 0 2.99 1.99 1.49 -1.,07 1.20005 -o.,24.46 0.0 0.0 35 .9286 -8.8460 0 .0 0.0 26 L 3 .76 0.18 013? 1.55 3.74 0 .8? -1.,74 1 .04845 -0.,24346 0.0 0.0 30 .4202 -8.8460 0 .0 0 27 LS 4 .00 0.21 0.32 1.21 4.66 0 .55 -2.,uO 0 .98B61 -o,,24346 0.0 0.0 28 .2459 -8 .8460 0.0 0.0- 26 SDH 4 .13 0 0.33 1.14 4.9* -n .01 -0. 0 .980?3 0,.11495 0.0 0 .0 27 .7747 1 .6494 0.0 0.0 29 •JDH 4 .?•! 0• 25 0-33 1.22 4.7R -0.57 1,56 1.01905 0,.47901 0.0 0 .0 28.6833 12.22t>6 0.0 0.0 Table- 5(continued) " " 35 LS 4,.51 28 0.,34 1. 4.06 -0. 84 1.38 1.,13tT7S 0.47901 0.,0 0,.0 31.6886 12. 2S66 0.o G.C, 31 30 4..51 o ?8 0.,34 1.57 4.,06 -0. 72 1.07 1.,13679 0.39115 0,,0 0 .0 31.6686 10. 9348 0.,0 0.0 "32" LS 4..76 0.30 0.,35 1.96 3.57 -0.96 0.94 1.,23293 0.39115 0.0 "0 .0 34,.3886 10. 9648 0.0 0.0 33 SAL 4,.76 0,,30 0.,35 1..98 3.,57 -0.,97 0.,96 1..23293 0.39956 0.0 0 .0 34 .3886 11.,1037 0.,0 0.0 ' "34 A' 4..89 0.,31 0.,J6 2.24 3.,J1 -0..V5 0,,89 1.,26908 0. 40971 0..0 0.0 35 .8839 11.,5443 0.,0 0.0 35 S3 4..89 0.,31 0..36 2.,24 3.,31 -0.,95 0.,89 1.,28908 0.40930 0..0 0 .0 35 .8839 11 =,5365 0,,0 0.0 36 3 5..03 0.,32 0,.37 2.,50 3.,07 -0.,90 0..81 1..34646 0.41573 0,.0 0 .0 37 .4298 11..8743 0,,0 0.0 37 B 5 .16 0..33 0 .37 2,,74 2.,86 -0..81 0. 1,,40462 0.41B39 0.0 0 .0 39 .0126 12..1023 0,.0 0.0 38 Sb 5 .16 0.,33 0,.37 2..74 2.,36 -0..61 0.[74 1..40462 0.41790 0.0 0 .0 39 .0128 12.,0954 0,.0 0.0 "39 5.30 0..33 0.39 2..95 2.,66 -0..70 0,.66 1..46298 0. 41676 0,.0 0 .0 40 .6178 12.,2136 0,.0 0.0 " io fl 5 .44 0,.34 0 .39 3..13 2..48 -0..5b 0,.59 1,.52106 0.,41183 0 .0 0 .0 42 .2311 12,.2209 0,..0 0.0 41 SS 5 .44 0,.31 0 .39 3,.13 2.,48 -0..56 0,.59 1,.52106 0..41126 0 .0 0 .0 42 .2311 .2129 0,.0 0.0 42 " B ' 5 .57 0.,35 0,.40 3,.27 2.,33 -0..39 0..52 1..57827 0.,40259 "0.0 0.0 43.8370 12,.1093 0,.0 0.6 43 B 5 .71 0,.35 0 .41 3,.36 2..19 -0..22 0,.44 1,.63414 0..39027 0 .0 0.0 45.4219 11,.8953 0,.0 0.0 44 SB 5.71 0,.35 0.41 3,.36 2,.19 -0,.22 0,.44 1,.63414 0,,38961 0.0 0 .0 45 .4219 11..8865 0.0 CO ri 5 .34 0,.36 0.42 3,.40 2;.06 -0.03 0.37 1..68809 0..37374 0 .0 0 .0 46 .9703 11 .5650 0..0 0.0 46 SAL • 5 .84 0, .42 3,.40 2 .00 -0.06 0.39 1.68809 0,.36946 0 .0 0.0 46 .9703 11 .7663 0 .0 0.0 47 LF 6 .01 0,.37 0 .43 3,.43 1,.96 -o.ii 0..30 1.75204 0,.38946 0.0 0.0 48 .9057 11 .7863 0.0 0.0 45 •JFH 6 .14 0,.37 0 .44 3,.36 1..96 0.66 -0,.24 1.77641 -0,.02160 0.0 0 .0 49 .6808 -0 .0267 0.0 0.0 •49 JIFH 6.27 0,,3H 0.45 3..09 2..09 1..35 -0,.60 1.74639 -0,.43193 0.0 0 .0 48 .8982 -11 .8433 0 .0 0.0 50 LS 6.52 0 .39 0 .47 2,.48 2 .53 1 .12 -0 .99 1.64021 -0,.43198 0 .0 0 .0 45 .9871 -11.6433 0.0 0.0 51 L .14 0 .45 0 .50 1,.43 4 .07 0 .56 -1 .48 .37121 -0,.43198 0 .0 0.0 38 .6123 -11.8433 0.0 0.0 ' 52 L 77.76 0 .53 0 .52 1 .10 6 .12 ~Q .01 -1 .9/ 1.10222 -0.43198 0 .0 0 31 .2374 -11 .8433 0 .0 o.o 53 LS 3.01 0 .57 0 .53 1 .16 7.24 -0 .24 -2.16 0.99603 -0.43198 0 .0 0 28.3264 -11 .3433 0.0 o.o POS S !3X 5.09805 THETA= 6.10017992 1 2.26343 &Y« 2.10328 TGAM 1.96020 0.0 j MAXIMA RETAXC 20) = 7.38207 XE1 48) 1.77641 BETAYC 1)* 7.44787 YE6JC 53) 0.0 MIMIVA cETAxC 52)* 1.09595 XtO=( 1)= 0.84166 BETAY< 21) = 0.89751 YEolC 53)= 0.0 CD^TRiajTinNS TO CHROMATICITY (D'JU/CDP/P)) FROM MAGNETS EXPLICITLY IN CHRX * -6 .4493, CHRY = -7.0719