JAERI-Conf 95-021

PB31
LOW-EMITTANCE SLOW EXTRACTION USING HALF-INTEGER RESONANCE
Yoshihiko SHOJI, Katsumi MARUTSUKA, Takeshi TOYAMA and Hikaru SATO
National Laboratory for High Energy Physics
1-1 Oho, Tsukuba-shi, Ibaraki-ken, 305, Japan

unstable regions. The tune is then approached slowly
to a half-integer value, and the stable region becomes
smaller and smaller as the tune approaches thishalf-integer, and the particles are ejected from themachine. This is the process of extraction.
Although particles with different momenta areusually extracted at the same time, they follow
different separatrices in phase space. The differencein the angle at the ESS is the dispersion angle of theextracted beam, which is responsible for the angle
divergence of the beam. One reason for thedisplacement is a difference in their tune, which
produces. a difference in the stable area in theseparatrix. Another reason is a displacement by thedispersion function. These effects are schematicallyshown in Figure 1. Although the first displacement
depends on the chromaticity and the second one doesnot. Therefore, when one choose· the chromaticity
properly, the separatrix arms of different momenta
overlap as shown in Figure 1. Actually in halfintegral resonant extraction, the separatrix arms do
not overlap, but only cross, because the separatrix
arms are not straight lines. However, this ispractically sufficient for efficient slow extraction.

INTRODUCTION
At the KEK-PS a resonant slow extraction using ahalf-integer resonance (2v h =15) has been inoperation. This article deals with a particular
problem concerned with the separatrix dependency
on chromaticity of resonant extraction. Thatdependency was first calculated by H. Bruck et a!. [1]and later by W. Hardt [2] for 1/3-integer resonant
the . outgoingthatshowed
Theyextraction.
separatrices of different momentum overlap under acertain condition (overlap condition). It is desireble
to obtain a separatrix branch with low divergence at
the first septum (usually ESS) position in order tominimize particle losses on it. Although the designof a resonant slow-extraction system is usually doneby numerical ray-tracing, analytical guide formulae
for a simple estimation of the parameters isdesirable. Unfortunately, the overlap condition has
never been calculated for half-integer resonant
extraction. This seems to be due to a difficulty intreating the parabolic form of the separatrix arm, toa straight separatrix. arm of the third-integer
resonance. In addition, twiss parameters (a., f3 and y)
introduced by Courant and Snyder [3], and thechromaticity go to infinity when the tune approaches
to a half-integer. The formulae given in this article
have been applied to parameter setting of resonant
slow extraction at the KEK-PS.
In half-integer resonant extraction at the KEKPS, one perturbing quadrupole magnet is inserted
into the lattice in order to produce a half-integer
stop-band. The total tune (v ) is quite close to a halfinteger (2v "' n). One octupole magnet is excited inorder to separate phase space into stable and
0. 01

Here, the twiss parameters and chromaticity (~)
of the unperturbed lattice are assumed to be known.
Since un unperturbed lattice has no perturbation'quadrupole, it also has no half-integer stop-band.
Using the formula calculated by T. Suzuki and S.
Kamada [4], the separatrix line in the unperturbed
normalized phase space IS expressed asyyy(a)III CALCULATION OF SEP ARATRIX

( b)

0.01

(c )

0.01

-0.01

-0.01

-0.01

-0.01

Figure 1 Scparatricies in unperturbed normalized phase space. The machine parameters of the KEK-PS areused. (a) Separatricies with different emittance: 0 and 3n mm mrad. (b) Displacement by the dispersion+0.4% and -0.4%). (c) The outgoing separatrix lines are crossing at the ESS(momentum displacement wasedge. One separatrix is of a particle with zero emittance and zero momentum displacement. Another separatrix
is of a particle with 3n mm mrad emittance and -0.27% momentum displacement.

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JAERI-Conf 95-021-X sin}! i +Y COS}li=tan(}! j-}1 i)/{ 1+( EB )/[Bcos 2(}1 j-}li)]) [Xcos}l i+ Ysin}l i]:!: V0/2[Bcos"(ll j-}1 i)+(E-B)] [(Xcos}li+ Ysin}l ;)2-E( E -B)/{D[Bcos2(ll j-}1 i)+( E-B )] ) ] .
(1)

X*=±vfW/B Xs,
Y*= ±VW/B Ys,
H'*= ±VWIB H',
H*=±VWIB H,h'*= !.:{2D/B h' and~* = ~/[Bcos 2 (}1 j-}1 i)] ,Coordinate (X, Y) is defined according to a wellknown form:

so that they contain neither B or D explicitly.
Equations (5) anct (7) become

X= x/{73Y = {73and[ x' +(f3/o. )x ] .

(2)

The two phases (ll i and ll j) are the unperturbed
betatron phase advance from the entrance of the ESS
to the octupole magnet and to the perturbation
quadrupole magnet, respectively. The strength ofthese perturbations are represented by D and B,
which are defined by
D = (1/1211)(ll. Bz"'f/Bp) p;2
B = (l/211)(ll. Bz'f/Bp) J3j.

and(4)

Since the unperturbed tune (v ) has almost a halfintegral value (n/2), E = (n/2)-v is very small.
We treat the case when the perturbed tune of aparticlewithzeromomentumdisplacement(dP/P = o=O) is just n/2, which means that the
separatrix has just zero stable area. However, aparticle with o "0 has a finite stable area (11 Emax)
because of the chromatic tune shift. We first
calculate the separatrix when o =0. In this case E= B
and the separatrix is expressed as

On the other hand, when o "0, E is writLen as E = B~o using unperturbed chromaticity. Therefore, theseparatrix is-X sin}! i +Y cos}! i=tan(}! j-}1 i)/ ( 1-~ o )/[Bcos2(ll j-}li)])[Xcos}l i+ Ysin}! i]:!: -(D/2[Bcos 2 (}1 j-}liH o]{(Xcos}l i+ Ysin}l i) 2+~o(B-~o )/{D[Bcos 2(ll i-lliHoJ)).
( 6)
The displacement due to the dispersion function (i")
and i") ') is taken into the calculation due to thedispersion in the unperturbed normalized phasespace (H and H'). Since equilibrium orbit isdisplaced by (Ho , H'o ), the X and Y of equation (6)should be replaced by X-Ho and Y-H'o, respectively.
We now define the Ys so that the (Xs, Ys) satisfy
equation (5) and (Xs, Ys+h'o.+h"o 2 + - -) satisfy
equation (6). Here, h' is the dispersion angle of theextracted beam in normalized phase space. Upon
subtracting equation (5), and omitting more than the2-nd order terms to o , we obtain

To simplify the equations, we rewrite them using

The sign of equations (8) is selected in order that
(X s, Ys) is on the outgoing separatrix arm, the stepsize of X* times X* should be positive. Only one oftwo solutions of equation (10) is correct; that is(12)(13)

Equation (11) is solved a straightforward mannerusing equation (12) as(h'*-H'*)(l-F)cos}l j +H*[sin}lj +Fcos}lj cot}li]= ~* {1+[Fcos}l j cos(}lj-2}1 i)/sin2}1 i-X*cos(}lj-}1 i)/sin}! i }/2) .

(14)

As has been explained, since the separatnc1es crosswhen h'*=O, setting h'* to 0 gives the crossing
condition of the separatricies at the ESS.
~ * {1+[Fcos}l j cos(}lj-2}1 i)/sin2}1 i-Xsin}lj+Ycos}lj = :!:V0/2B (Xcos}li+Ysin}li) 2 .(5)Hsin}l jj+(h'-H')cos}lj= {W2Bcos 2(}1 j-}1 ;)]) [Xssin(}l j-2}1 i)+ Yscos(}l j-2}1 i)]:!: (20/B [Xscos}l ii+ Y ssin}l i][-Hcos}l i +(h'-H')sin}l i]:!:y'B/(2D)WBcos 2(}lj-}li)].
(7)2( -X* sin}! j +Y*cos}lj) = (X*cos}l i + Y*sin}l ;)2,
( 1 0)
H*sin}l j +(h'*-H'*)cos}l j+(X*cos}l i + Y*sin}li)[H*cos}l i +(H'*-h'*)sin}l i]= ~*{1+[X*sin(}lj-2}li)+Y*cos(}lj-2}li)]/2) .
(11)

Y*sin2}1 i = Fcos}lj-X*sin}l i cos}lihere
F = 1-[1-2X*sin}l; cos(}! j-}1 i)/cos2}1 j] 1/2 .

(3)

(8)
(9)

-X*cos(}l j-}1 i)/sin ll i ]/2) = (H*sin}lj-H'*cos}l j)+F[cos}l j /sin}li][H*cos}l i +H'*sin}l i]( 15)This is one goal of this article. However, in thecalculations some small effects are ignored, such as
the momentum dependence of ll i, }lj, o., p, y, i"), i") ', Dand B, and the change of ~, i") and i") ' due to theperturbation quadrupole, and a higher order effect too. Although it is possible to use them .in thecalculation,
theresultantequation
istoocomplicated. A computer-tracking simulation would
be more suitable to obtain more accurate solution. Itwould be helpful if we have a more simple, but lessaccurate, formula for the crossing condition. In thelimit X*=O, which means that all separatricies pass
the equilibrium center, equation (15) becomes much
more simple as~* = H*sin}lj-H'*cos}lj(16)

We next calculate the chromaticity dependence of
the angle divergence. The momentum spread of theextracted beam at one instance (o max) is determinedby the emittance and chromaticity. The angle
divergence at the ESS edge (fl. Y in normalized phasespace) is the product o max h'. Since fl. Y is notconstant during the extraction, we calculate thespread at the middle, where the fl. Y is maximum. Theomax is determined from the emittance (11 Emax,maximum Courant Snyder invariant), which is-336-JAERI-Conf 95-021

11Emax=( 4{2/3)[ E ( E-B) ]312j ( D[Bcos2(}1 j-}1 i)+( EB)j2}"' (4{2/3)[1~*1omax] 3 12 [B/Dicos 2(}1 j-}li)l]. (17)Therefore,om ax"' [ (3/4 {2)( 11 EmaxD/B )Ieos( }-I j- }1 i)IJ213jl~ *I ( 18)
However, when the right side of the above equation islarger than the full momentum spread of thecirculating beam ([t.P/P]max), omax is equal to[t.P/P]max. When omax is smaller than [t. P/P]max, themomentum center of the extracted beam shifts fromthe beginning of the extraction to the end (when thecirculating beam is not accelerated during slow
extraction).

have given the minimum. The shifts in the beam
position from the start of extraction to the end was areflection of the shift in the central momentum of theextracted beam. When the chromaticity was zero, thebeam was split at the start of extraction because of
the non-linear chromaticity. This curious beam
behavior was reproduced in a computer-tracking
simulation.
angle divergence (rad)0.00120.00
0.000

III APPLICATION TO THE KEK-PS

The parameters of the KEK-PS are:Figure 2 Maximum angle divergence at the ESS as afunction of the horizontal chromaticity (~).

J3 = 15.8m at the ESS

0. = 2.20;}lj = (1 05/28)11;}1 i = -(7 5/28) 11 ;xs = 32mml") = 1.86 m

B = 0.051
D = 100/ml)'

CONCLUSION

IV= -0.1076 mrad.

Inserting these parameters into equation (15) gives
This means that the angle~ *=24, i.e. ~ =0.88.
divergence is zero when the chromaticity is set at thesmall positive value: 0.88. When we use the simple
equation (16) the result is ~=0.41.
The maximum angle divergence t. Y max/]!l3 is
calculated using the parameters of the KEK-PS at thetop-energy: 11 Emax = 311 mm.mrad and [t. P/P]max =0.8%. The result is shown in Figure 2.
The chromaticity dependence of the extracted
beam size was observed at the KEK-PS while
extracting 3.5GeV protons. The extracted beamprofile at several timings during the slow extraction
was measured by the SWIC (Segmented WireIonization Chamber) at the down stream of the last
magnetic septum (betatron phase advance from theESS was about (3/4)11 .) The horizontal beam profiles
are shown in Figure 3. Although the beam width wassmaller at a small positive chromaticity, themeasurement missed the chromaticity which would

The crossing condition of the separatrix arms ofanalyticallywasresonancehalf-integral
thecalculated.
This has already been utilized at the KEK-PS
during the extraction of protons and a light-ion beamwith smaller rigidity. Unfortunately, at the time of
top-rigidity the power supply of the chromaticity
control sextupoles is not powerful enough to overcancel the negative chromaticity to positive.
However, below lOGeV (proton) this condition can berealized, and produced a dramatic improvement in
the extraction efficiency and emittance of theextracted beam.

REFERENCES[1] H. Bruck et al, Proc. of HEACC'74, p471.
[2] W. Hardt, PS/DL/LEAR Note 81-6, CERN, 1981.
[3] Courant and Snyder, Ann. Phys. l. 1(1958)[4] T. Suzuki and S. Kamada, KEK-76-7, 1976.

-'_______ /....___~-----..... ....................
;---------~~~-_______.,---·,_/',~/o-...../'___.,..-'--------------

f\J_____...--...._----------f'-'=-~............... ....--~'/'---./'~\'_/['\!

''I)tIJ!

/I~_).!"--')=5:,\Jl\;"1\---'7\IVI_/\/..\!

I!

_/
_/
_/[:'\.LD,/~~:'-,\I;,\\II

i:\---"'i:;Horizontal beam profile of the extracted beam at the SWIC. The chromaticity was -7, -3.5, 0, +3.5, + 7
Figure 3from the left. The time from the top trace to the bottom trace was about 2 seconds. The full-scale of thehorizontal axis is about 80mm.

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