Spin flip by adiabatic passage of depolarizing resonances in synchrotrons

N U C L E A R I N S T R U M E N T S AND METHODS

165 (1979) 5 5 3 - 5 5 9 ,

(~) N O R T H - H O L L A N D P U B L I S H I N G CO

SPIN FLIP BY ADIABATIC PASSAGE OF DEPOLARIZING RESONANCES IN SYNCHROTRONS MASAAKI KOBAYASHI

National Laboratory for High Energy Physics, Oho-macht, Tsukuba-gun, lbarakt, 300-32, Japan Recewed 29 January 1979 and m rewsed form 18 May 1979 If a strong depolarizing resonance is crossed slowly as m many strongly focusing proton (or electron) synchrotons, the beam polarlzaUon is expected to fhp completely ff the synchrotron oscillation ~s neglected An mtumve picture for the spin fhp ts given by using the Fro~ssart-Stora equation As the resonance approaches, the phase of spin with respect to the oscillating honzontal field ts "locked" m a certain stable region An oscdlatlon grows m the vertical spin component sz, and sz decreases as a whole After the resonance, the oscillation ts gradually damped, pulhng down sz to the inverse sign as before Weak and t o o strong resonances, and the influence of synchrotron oscillation are also discussed

1. Introduction It is well known that polartzed protons (or electrons) in synchrotrons lose the polarization as they cross the intrinsic resonance I) The resonance condition is given by

yG= kN+vz,

(k=0,+l,

+2,

),

(1)

where ? = ( 1 - / / 2 ) -1/2 is the conventional Lorentz factor and G =g/2-1 is the anomalous magnetic m o m e n t in units of eM2mc (m the particle mass) Khoe et al 2) gave a formula for the depolarization due to the resonance The change of polarization P 0 - P is, as long as it is small, given by (Co - e)/eo = 2 [1 - exp ( - x)-I, X = In(1 +yG)r]Z/GAy

(2)

where zl~ is the increment of ~, per revolution The quantity r, ref 2, is the ratio of clockwise rotating horizontal field with the angular velocity of ~G = k N _ v~ to the guiding vertical field (BtL)) In strongly focusing synchrotrons, r IS much larger than in weakly focusing ones The depolarization is sometimes large For example in the KEK 500 MeV booster proton synchrotron (PS), X is 125 for the resonance at ~,G = v, = 2 25 if r is taken to be 0 004 It has been predicted3-5), based on large values of X, that spins will completely flip by slowly crossing a strong resonance The mechanism of spin flip, however, does not seem clear enough A few questions arise Is the spin flip also true for resonances which are too strong9 Does the once flipped spin stay stable throughout the resonance without oscll-

lating many times 9 How does the spin as a classical top behave during the resonance 9, etc Moreover, an expenment 6) at ZGS to venfy the prediction of complete spin flip showed only a partial flip The authors attributes the incomplete flip to synchrotron oscillations How can one explain the incomplete spin flip in terms of synchrotron oscillations 9 In this paper we wish to answer these questions To this aim, an intuitive picture for the spin flip is worked out, based on the Froissart-Stora spin equation m the case of no synchrotron oscillation The effect of synchrotron oscillation is qualitatively discussed m the last section The k = 0 resonance is considered for simplicity, though the other resonances can be treated in a similar way A computation for the KEK booster PS is also presented as an example 2. Equation of spin motion We choose the turning rest flame of the particle whose origin lies on the equilibrium orbit Its x-y plane rotates around the guiding field with the z

9.

...y

Part,cles (e.~,O) x Fig 1 The turning rest frame of the partmles, (xyz)

554

M KOBAYASHI

Larmor angular velocity of e / m ? The z axis is always vertical, the x axis radial and the y ax~s on the equlhbrmm Orbit (see fig 1) Time t IS contracted with respect to the laboratory time t (L) by ?-~ The present frame is natural and convenient as ~t moves with the particles, yet maintaining the clear meaning of the three axes Strmght sections are neglected because the fringe field IS not responsible for the k = 0 resonance In the laboratory frame, the spin vector s, normalized to the unit length, satisfies the Fro~ssart-Stora equation 3)

d$ e dt (L) -- m~' s x r(1 + G)BI~) + (1 +yG)B(L)],

oo,(f° t

s

=s,

+ (8)

sy

sj_ sin

(L ~%(0dt + a(t) )

It is convenient to see the phase of spin vector with respect to the oscillating field We rewrite (8)

as si = s± sin

(;o

)

COm(t)dt+ a'(t) ,

'(t) = a(t) + 2(t/2--tres)t,

(3) = a~-am¢

(9)

f l dy'~ GAy :eBz~ 2 "~ c°~sl---=-!\y dt/~es = 2rr Ik--m-"}

where BllL) (or B~L)) Is the laboratory field parallel (or perpendicular) to the orbtt Approximating as ~L) N B~L) and B}L)--BxL>+B~L), and subtracting from (3) a term ( e / m ? ) s × (~L)), we have the equation m the turning rest system as

Subsmuuon of (6) and (9) m (4) gwes the follow-

ds/dt = Ks x B ,

l (4)

B~ = (1 +TG)B~L)/(1 + G), B~ = yGBf~L)/(1+

(;o)

mm(t)dt ,

(B~° > 0),

(5)

oscillating with the angular velocity 09m = (kN-+-vz)eB~L)/rn Writing (.0 m -~- O.)p -~-- (.Ore s at the resonance, we have in the vmlmty of the resonance

O)m(t)

=

(/)re s +

dB~L)'] arn ----"Ogres ~zzL, dS ./res

(6)

(1

=

(-Ores +

ap(t--tres),

( i dy + B1) ~ dB(zL)') __ (.Ore s

dt

(10)

I

da' = ;t(t-t~,~) - 1 :xs, B°.~

\-ZU) × X {COS[2Ogr.st + a m ( t - 2 t r J t + a ' ]

(7)

dt /r*s

Because usually Bx ~ B z , Sx and sy can be approximately written as

- cosa'}

Though numerical integration of the above equations gives the spin motion, an approximate spin motion can be seen more simply by averaging the above equations dunng a period of oscillating field Th~s ~s acceptable because durmg most of the t~me sz and 7' change slowly Observing the spin each Ume the phase of the oscillating field crosses multiples of 2 zc, we have

dsz d0c'

[dr =

The spin precesses around B with the angular velocity

ap =

+ a'] -- sin a'}

slnA

-½•s.t.B °

dt =

am(t--tr,~),

COp(t) = TGeB~L)/m

dsz = ½xs±B o {sin [2cO,ost + am(t--2tr~)t + dt

d,

O),

where x = g e / 2 m B is an effective field m the turning rest frame ParUcles are affected by a horizontal field B~ = - B ~ sm

mg

sln(a'+A)

d(1 +d/2rr) '~'(t--tres)

+

I:KszBO~

' Sill]

(11) ,

~ ~ - - ~ }/I (1-~-/2~) cos (a + A),

where A = ~z2(t-t~s)/Wm xs usually much smaller than unity 3. Slow motion of spin The spin motion described by eq (10) is determined by two mechanisms The first mechamsm comes from a difference between the particle energy and the resonance energy, and corresponds to the ;t term In do:'/dt Before

SPIN F L I P IN S Y N C H R O T R O N S

o('= 0

A3

Y

555 Z

1

Fig 2 If B x ts nonzero, the spin vector moves from A 0 to A 1 after a penod o f oscfllatmg field even at the resonance point (see text)

Fig 3 A ~chematlc l~lcture for spm Nip A m o v e m e n t of the crest of the spin vector is shown as observed each t~me the oscillating field B x takes its m a x i m u m m the negative (see text)

the resonance, the precession is retarded with respect to the field osctllatlon, while after the resonance it advances The second mechanism comes from the honzontal field and corresponds to the Bx term in d~'/dt Even at the resonance point (corn = Ogp), the spin vector does not necessanly come back to the startlng posttlon after a precesston pertod if nonzero honzontal field is present Referring to fig 2 the crest S o f the spm vector, when projected on the horizontal plane, sits on A0 at a certain time Now that Bx < 0 during half a period, S rotates roughly around a point N which is separated from the origin by an order of magnitude of B~/B~, reachmg Bo instead o f B~ In another half period, the hortzontal field is mverse In sign S rotates by 7r around the pomt P, which is the mirror tmage of N with respect to the origin Consequently, the spin vector moves from A0 to A~ d u n n g a period It moves to A2 at another period later, and so on

quently, around the beginning of the resonance region, ~' may be assumed to he between - r r / 2 and re/2 As the Bx term is approximately proportional to cos ~', it is difficult for ~' to cross +__rr/2 to go outside ~' is, therefore, typmally confined between - r e / 2 and 7r/2 throughout the resonance region One can see from eq (11) a schematic m o v e m e n t o f spin as sketched in fig 3 The spin vector is represented by its crest at the sampling time when the phase of oscillating field crosses multiples o f 2 7r M o v e m e n t from A to D constitutes a period of spin oscillation At the point A, ad takes its m a x i m u m Both s" and ~' decrease to reach B, at which sz starts to increase The Bx term soon supercedes the ;t term (C), and a? then starts to increase As sz contmuously increases, ~' increases raptdly to run up to D at whsch ~t' takes its m a x i m u m again F r o m the above, we may have a typical OScllla1

4. M e c h a n i s m

of s p i n f l i p

F r o m eq (11) one sees that when the resonance ts far away ~' changes raptdly, causing no stgntficant depolartzation W h e n , on the contrary, the resonance approaches, ~' changes slowly The B~ term is important here Let us call this region the resonance regton W h e n the resonance approaches the situation in which both 2 and Bx terms are comparable with each other, da(/dt ts significantly reduced by a cancellation between both terms tf ~' lies between - n / 2 and zt/2 If ~' lies outstde the above region, however, the two terms s u m wsth the same sign and will soon drive ac' mto the above region Conse-

N U3 0

~~t :i

res

tl t2 i! ~ E

(a) --,-

-1 1

o

(b) b

Fig 4 (a), sz decreases as tt oscillates, (b), is not the case (see text), tr¢s shows the resonance point

556

M KOBAYASHI

tlon pattern o f s~ as sketched m fig 4(a) The oscillation grows as the resonance approaches and is damped as the resonance goes away A spm flip Is the net effect Let us conmder below why Sz osctllates as thts pattern shows First, why does the osctllatlon grow as the resonance approaches 9 This is because the amphtude is roughly proportional to the OSclllatton pertod [see eq (11)], which increases as already explamed above Second, why does the m a x i m u m of oscdlatlon s~ decrease as seen in fig 4 ( b ) 9 This pumpmg-down mechanism is due to the existence of a slowly changmg bias 2(t-t~,s) m dod/dt Equation (11) shows that as the resonance approaches closer, ~' changes its s~gn at smaller s, The m a x t m u m of s~ ts then soon reached while s~ ~s sttll smaller than before W e have seen above that as the resonance approaches, the osctllatton grows in s~ and the m a x i m u m of s~ decreases After the resonance point, the s~tuatlon ~s inverse to the previous one The oscillation becomes fast, thereby damping the amplitude The m i n i m u m of Sz goes down to - 1

Weak or too strong resonances If a weak resonance approaches, sz experiences a small oscillation, gradually decreasing as a whole Because Sz does not come close to - 1 , the Bx term m d~'/dt is much smaller than the 2 term soon after the resonance point Consequently, 7' tends to increase monotonously The depolarlzmg m e c h a m s m soon

ceases to work A certain degree of depolartzatlon is the net result as sketched m fig 5(a) If the resonance is too strong, a( ~s dominantly governed by the Bx term rather than the 2 term The oscfllatton amphtude o f s, is already large long before the resonance pomt The pumping-down mechanism, whmh draws s~ as a whole down to - 1 , is at a first glance hidden m the s|gntficant oscdlauon, though it is still working At a certain while after the resonance point, the ). term grows comparable with the Bx term The oscillation in s, is gradually damped and the pumping-down mechanism becomes dominant The net result will be the complete flip as sketched m fig 5(b)

Width of resonance The resonance mechamsm will work as long as the 2 term is as small as the Bx term m dod/dt If the resonance width ~L~ tS defined as the laboratory t~me necessary for sz to change from + 0 9 to - 0 9, the above condmon gives F(L) ,,~ s r(l + yG)/G2 rcs - - 2 COre

(12)

Writing ",ev~(L)for the pertod of revolution, we have roughly -,(L)

~___2~, 2

(I + yG'~ (B~)'~

(13)

~rev

which ts in reasonable agreement w~th the computations

Condttton for complete spm flip The condition X [given by eq (2)-] >>1 i

fires

(a) Weak Resonance

rl"a--~

" I i

-1

ltln (b) UU

Too Strong Resonance

-1 F~g 5 T~me dependence of sz IS schemattcally shown for (a), weak resonances, and (b), too strong resonances tres shows the resonance point

(14)

can be quahtatlvely derwed as follows the osclllaUon horizontal field can be decomposed into two rotating components clockwise and antlclockwlse, of which only the clockwtse component ~s ~mportant In the turning rest frame the spin rotates around the clockwise rotating field b+ wtth an angular velocity of e(1 + 7G)b+/m The horizontal field may work on the spin coherently roughly during a period z' m which the phase difference between the rotating field and the spin precession reaches 2 zc We have z' ,-~ (81r/2) 1/2 = 4 7rm/[eB~L)(AyG)1/2]

(15)

A reqmrement that the spin rotation around the honzontal field dunng r ' / 2 ts m u c h larger than 7r reads now hke eel (14)

557

SPIN FLIP ]N SYNCHROTRONS

[(fl2/B(zL)) dB~)ldt'],~, =

65 2 s-'

5. Computation for the KEK booster PS

[(11~) dy/dt],,s =

Machine parameters are as follows vertical tune v ~ = 2 2 5 and radms o f c u r v a t u r e = 3 3 m The mtnns~c resonance occurs at y~, = vz/G = 1 255 (the klneUc energy = 239 MeV) T h e period o f betatron oscdlatlon at the resonance is z = 4 05 × l0 -s s (cO,¢s = 1 55 × 10 s s 1) m the turning rest frame T h e vertical oscdlat~on a m p h t u d e of 1 c m corresponds to B~L)/B~L)= 2r = 0 008 T h e energy gain o f 7 keV per turn is assumed, giving

Figure 6 shows spin flips of a particle for various strengths of horizontal field The oscdlatlon in s, increases on approaching the resonance and is d a m p e d on leaving it, with a resultant fl~p The phase o~' is almost constant m the resonance regton T h e vicinity of the resonance is shown in more detail in fig 7 Dependences on Ume of s,, ~' and z l ~ ' = the increment of o~' d u n n g a period of-field oscillation are presented, confirming the p~cture given in the previous section C o m p u t a t i o n is m a d e by numerically integrating eq (10)

L

6. C o n c l u s i o n s and discussions A spin flip IS shown schematically In fig 3 The spin vector ~s represented by its crest at the

~o tres

~B(L) - 0

1

0004

I

a

AT - 7 keY I

-1

0

1

2

. I

..

3

I

4

5

6

tres~

Nm / 104

o

,,

Nm 1103

~B~L} -'0 0 0 4 AT

30

28

'

,/

"

-1

. -8273 9~

3z;

-oooo,

i/

- 7 keV

X0

32

AT-?key

:a'={)n x Inte

25-32"55~i~ ^^1 ]

0

1

2

3

L

[ I

~tBx term i

<3 _0 2 ~ - - - ~ - 4

5

6

~ X

ter~

7 i

Nm / 104

~ =0 004 ~..~(L ) T .t~,~ 1

1F

I I

32

~o

0

AT = 7 keV

BIL) ~ / AT = 7 keV-]

L 2

,

C

i

,L_ 5

3 N m/10

.l

.

7

~

Fig 6 Spin flip o f a pamcle crossing the yG = o z resonance m the K E K booster PS for various strengths of honzontal field

/v.(= f: ~°~,,d,/2=) is the

number

of oscillation m the horizontal field At (a) B(xL)/B~ L) = 0 0004, (b) 0 004, a n d (c) 0 01 T h e arrow indicates the resonance point

N m = 0 (t = 0), s z = 1

34

.

.

.

.

4 -32`03 - 8 = 3 , 3 ~ i - - -

.

.

- -

I

} o ~ ° 2 B,,erm ] ~: -02

/oC=2nx Integer

^/~AA1 , I - - ' l . . . . . Y ...... , _~

.............

i

[

Fig 7 In the vlc[mty o f resonance, Sz, at' and zlat' = increment o f at' d u n n g a p e n o d of oscdlatmg field are given N m is the n u m b e r o f oscillation m the honzontal field A t N m = 0 (t = 0), s z = 1 and a t ' = 0 (a)B~L)/B~ L) = 0 0004 corresponds to fig 6(a) and (b) 0 004 to fig 6(b)

558

M

KOBAYASH]

samphng t~me when the phase o f the oscillating horizontal field crosses multiples of 2~r Long before the resonance, the spin vector rotates around the z ax~s maintaining the m~tml polarization o f s, = 1 As the resonance approaches, the spin vector ~s driven into a stable regzon o f ~z' between - z ~ / 2 and z~/2 The oscillation ~s slowed down, growing m the amphtude for s: and pumping-down s: as a whole After the resonance point, the s~tuat~on ~s the reverse of the earher case The oscillation becomes fast again, damping the amphtude for s, and pulhng down s, as a whole to - 1 The resonance regzon ~s soon over and the spin vector leaves the stable region o f o~', begmmng to rotate again around the z ax~s maintaining the final polarization o f - 1 Assuming for the 239 MeV resonance m the K E K booster PS a beam o f 1 cm radms w~th a uniform d~stnbut~on m the phase space, we find that as much as 98% of particles will fl~p the spm The numerical calculation of depolarization ~s compared m fig 8 w~th the analytical formula (2) as a check The agreement between both ~s good up to large depolarization Even at too strong resonances, the spin ~s expected to fl~p as a net effect after a s~gmficant oscdlat~on Such strong resonances, however, seem ~mpract~cal in order to achieve complete spin fl~p It ~s only the phase relation between the spin vector and oscdlatmg field that keeps the s~gn of m~tml polarization m m e m o r y and ensures the correct sign of the final polarization Any small d~sturbance during the resonance may destroy the m e m o r y and may gwe a wrong s~gn to the final polarization

Effects of synchrotron oscdlatton Up to this point the particle energy has been approximated to increase hnearly with time Synchrotron oscillation superposes a smuso~dal variation on ~t Even ff the rf phase ~s always posltwe, the crossing speed across the resonance ~s increased by the existence of synchrotron oscfllaUon for some (typically half) particles The resonance strength for those particles ~s reduced and the spin fl~p may become incomplete If the rf phase oscillates between positive and negatwe, another effect of synchrotron oscdlatlon may emerge because a particle may cross the same resonance energy more than once The crossing occurs at odd times Is ~t possible to approximate the muluple crossing by a single crossmg by cancelhng out the residual even t~mes of crossmg 9 In order to see the influence of multiple crossing, we have traced the spin motion for the cases as shown m fig 9 Referring to the three t~mes o f crossing (b in the fig ), the energy increases before the timing to, decreases between to and tl and increase again afterwards Both Ummgs to and tl are vaned with IA~,I fixed The polarization decreases at the first crossing, recovers more or less at the second crossing and m many cases decreases again at the last crossing If the resonance ~s strong, the final polarization

t°-A//

225 t

,

"~/Vz'LINE t

_ /

~

= 000Z,

P,n,l,aL=I 0

,% -,

i go x-

-

i

-1 I

i

i

i

i

I

I

i

I

i

I

i

h

i

i

I

i

i

UI

Ftg 8 Final pelanzatlon for the yG = v z resonance m the KEK booster PS The mltml l~lanzatlon ts + 1 m the dlrecUon of synchrotron field The sohd curve shows eq (2) Sohd circles gwe numerical calculation w~th z t ) , = 7 k e V / t u r n and 2 r = B ~ L ) I B(zL) = 0 0001-- 0 001 Open ctrcles give B ( L ) / B (zL ) = o o o o 4 3 x and zl}, = 1 2 - 2 0 keV/turn

1

2

3

4

5

6

Nm/10 t'

Ftg 9 Effects of energy oscillation (the upper graph) on the spm motion (the lower graph) Nm IS the number of osctllatton ,n the horizontal field (a) has no energy oscdlatlon and corresponds to fig 6(b) (b) - (d) simulate various energy oscillations In the lower graph, only the average of oscfllatmg spin motions ~s shown except for the case (b)

SPIN FLIP IN S Y N C H R O T R O N S

shows a complete fhp If the resonance ~s medium strong or weak, the final polarization depends much on the timings to and t~, being mostly smaller m its absolute value than for a single crossing The above result may be interpreted m therms of "phase lock" of a( within a stable region If the multiple crossing occurs for strong resonances w~thln a duration of "phase lock", a depolarization and the subsequent recovery may cancel each other and the multiple crossing may be approximated by a single crossing If the second crossing occurs, for medium strong or weak resonances, when the "phase lock" due to the first crossing is complete, the phase ~z' is not yet prepared for another period of the proper depolarization process The effect of the second crossing may be various depending on the phase ~' The final polarization after the multiple crossing may then be different from that for a single crosslng From the above two effects of synchrotron oscillation, one may expect a reduction in the resonance strength If the resonance is strong enough, the influence of synchrotron oscillation will be small Whde the synchrotron oscillation is widely different from accelerator to accelerator, the KEK booster PS has the following parameters at the resonance synchronous phase N30 °, the stability region = - 3 0 ° - 9 0 ° and frequency N 10 kHz As the maximum accelerating voltage IS 16 kV, the local crosslng speed across the resonance is 2 3 times larger at the maximum than the average one From a computation of depolarization as a function of resonance strength, the effect on the complete spin fl~p is smaller than a few percent We can moreover estimate that less than 10% of particles may cross the resonance three times The effect of the final

559

polarization will be as small as 0 05 As a result the spin flip may be complete by more than 90% In some cases effects of synchrotron oscillation may be more serious One of solutions wdl be to cut the longitudinal beam phase space so that most of particles he always in the acceleration phase Reduction of the InJected beam in the longitudinal phase space is also useful Charge exchange injection of the negative hydrogen beam will meet this arm will Also, in order to study the effects of synchrotron oscillation, development of polarlmeters for the c~rculatlng beam, as pointed out in the polarized beam workshop at Ann ArborT), is indeed reqmred The author appreciates Drs K Monmoto, H Kobayakawa, S Takeda, T Suzuki, S Hlramatsu and Y Mort for helpful and enhghtenmg discussions He is also thankful to Dr D Mohl for critical reading of manuscript References l) E D Courant, BNL Report, EDC-45 (1962) and refs cited thereto 2) T Khoe, R L Kustom, R L Martin, E F Parker, C W Potts, L G Ratner, R E Tlmm, A D Krtsch, J B Roberts and J R O'Fallon, Panicle Accelerators 6 (1975) 213, T K Khoe, KEK Report KEK-73-8 (1973) 3) M Fro,ssart and R Stora, Nucl Instr and Meth 7 (1960) 297 4) L Teng, NAL Note FN 267 (1974) 5) J Faure, A Htla,re and R Vlenet, Particle Accelerators 3 (1972) 225 6) y Cho, R L Martm, E F Parker, C W Potts, L G Ratner, J Gareytte, C Jhonson, P Leferre, D Mohl and A D Knsch, 1976 Conf on Htgh energy phystcs wtth polartzed beams and targets, ANL (1976) p 396 7) Htgher energy polartzed proton beams (Ann Arbor, 1977), eds, A D Knsch and A J Salthouse, AlP Conf Proc No 42