Transparent Spin Resonance Crossing in

Accelerators

A.M. Kondratenko (1), M.A. Kondratenko (1)and Yu. N. Filatov (2)

(1) GOO “Zaryad”, Novosibirsk(2) JINR, Dubna

nr

0Sr

J

2πνΨ =

π2Sr

[ ] ,dS W Sdθ

= ×r

rr

Spin Motion at Circular Accelerator

In ideal accelerator

Thomas-BMT equation

,zen rr= Gν γ=

( ) ( )iiii InIn Ψ=+Ψ+ ,,2,,2 θππθ rr

[Ya.S. Derbenev, A.M. Kondratenko, A.N. Skrinsky (1970)] :

spin precession tune

Spin vector rotate around n-axis:nSnSJSnJS rrrrrrr

⊥⋅=+⋅= ⊥⊥ ,,

( ) ( ) −=+ θπθ nn rr 2

ν −

periodical axis of precession

020 || SSnSrrrr

=⇒ πIf

( )0 2 0 2, , 2S n S n S Sπ π πν⊥ ⇒ ⊥ ∠ = Ψ =r r r rr r

If

The spin equilibrium closed orbit

Not equilibrium orbit

spin deviation

−−= 2/)2(gG gyromagnetic anomaly

−Δnv

−Δν

Spin Adiabatic Invariant−⋅= nSJ rr

−Δ= Wwrr

spread of n-axisesspread of spin tune

⇒

−=Π Srr

vector of polarization,

−Π−=r

1D degree of depolarization

nJSnJ rrrr=+=Π ⊥

−θ particle azimuth

−ΨiiI , act-phase variables of betatron motion

Spin Motion Near Single Spin Resonance

Field in Resonance coordinate system,which rotates around vertical axis with resonance frequency, is

k z kh e wε= +r r r

is resonance strengthkw

Near coherent (imperfection) resonance( )θθν kekeweW yxkz sincos0

rrrr++=

( )θθεε

ε kekew

wew

n yx

kk

kz

kk

k sincos2222

rrrr+

++

+=

22kk wk ++= εν

Dependence of spin tune ν on Gγ in accelerator with and without resonance.

12222lrr

rr

k

kz

k wwe

whhn

++

+==

εkεk

εkis spin tuneis resonance detunekk ννε −= 0

22kk w+= εν

Hence, spin precession axis

In this system, spin tune

Spin precession near the spin resonance

when222 or kk

kk

k wddh

dhd ε

θε

θ+<<<<

r

Spin adiabatic invariant (SAI) is nSJ rr

⋅=

−Sr

blue arrows−nr red arrows

−effkε effective resonance

region

Adiabatic crossing( )beforeafterkk JJw =<<′ 2ε

Intermediate crossing( )beforeafterkk JJw ≠′ 2~ε

Fast crossing( )beforeafterkk JJw −→>>′ 2ε

constJ =

Generalized F.S. formula

M.Froissart, R.Stora, 1960

iz

fz SS ⎥

⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛−= 1

2æexp2

2π

normalized resonance strength

( ) −∞−= ziz SS initial vertical spin component

( ) −∞+= zfz SS final vertical spin component

−′

=εkwæ

if JJ ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

2æexp21

2π

Generalized F.S. formula

( ) ii i zJ S n Sθ= ⋅ ⇒ −

r r

( ) ff f zJ S n Sθ= ⋅ ⇒

r r

SAI magnitude becomes equal to vertical spin component magnitude only in the limit |θ |⇒ ∞:

In adiabatic region vertical spin component is 22cos

kk

kz

wJJS

+==

εεξ

Single crossing of the spin resonance

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛== +

2

1,

χ

χχχσχ rr

S

Spinor representation

The case of

( )χσθχ Widd rr

⋅−=2

−= ),,( zyx σσσσr Pauli matrices

The field coordinate system[ ] kk hhneene /;; 32221

rrrlrrr

lrr

===×=

⎭⎬⎫

⎩⎨⎧ ′

= kk

kk hhwW ;;0 2

εr

( ) ( ) ( ),0χθχ Ψ= zM0=′kε

0≠′kε

constJ =

( ) ⎟⎠⎞

⎜⎝⎛ Ψ−=Ψ zziM σ2

exp

,,0

0

ϕθϕθθ

θ

+=Ψ+=Ψ ∫∫f

i

dhdh kfki( ) ( ) ( )( ) ( ) ( ) ( )izyfzif

iiff

MMMM

M

ΨΨ=

=

αθθ

θχθθθχ

2,

,,

( ) ,2

exp ⎟⎠⎞

⎜⎝⎛−= yy iM σαα

( )ε

πα′

=⎟⎟⎠

⎞⎜⎜⎝

⎛−= kwæ,

4æexpsin

2

⎟⎟⎠

⎞⎜⎜⎝

⎛Γ−+−=

4æarg

4æln

4æ

4

222 ie

πϕ

( ) ⇒−= if JMJ 21221

The case of

∫=Ψθ

θ0

dhk

( ) ⇒−= if JMJ 21221

Generalized FS formula

( )−Γ x Gamma functioniif JJJ ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−==

2æexp212cos

2πα

Single crossing of the spin resonance

0.18æ =

66.0æ =

1.39æ = 1.39æ =

66.0æ =

0.18æ =

Two times crossing of the spin resonance

( ) ( ) ( ) ( ) ( ) ( )izyabzyfzif MMMMMM ΨΨ−Ψ= αβθθ 22,

The spinor transformation matrix is

( ) ,4æexpsin

2

⎟⎟⎠

⎞⎜⎜⎝

⎛−= aπα ( ) ,

4æexpsin

2

⎟⎟⎠

⎞⎜⎜⎝

⎛−= bπβ

ba

b

akab dh ϕϕθ ++=Ψ ∫a

a

ki

i

dh ϕθθ

+=Ψ ∫ bb

kf

f

dh ϕθθ

+=Ψ ∫

The condition of depolarization compensation (2 times crossing)

The SAI keeps sign

mM ab πβα 2,012 =Ψ=⇒=

( )if JJ =The case of SAI-flip

ππβπα +=Ψ−=⇒= mM ab 2,2

011

( )if JJ −=

Comparison with Results of A.W. Chao

“Spin Echo in Synchrotrons”, 2005Condition when spin echo is not present after two spin detuning jumps:

mabhk π2)( =−

This result of A..Chao is similar to our result for the case

mabhM

kab

ba

πϕϕ

2)(0if and012

=−=Ψ⇒=+=

Three times crossing of the spin resonance

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )izyabzybczyfzif MMMMMMMM ΨΨ−ΨΨ= αβγθθ 222,

The spinor transformation matrix is

( ) ,4æexpsin

2

⎟⎟⎠

⎞⎜⎜⎝

⎛−= aπα ( ) ,

4æexpsin

2

⎟⎟⎠

⎞⎜⎜⎝

⎛−= bπβ

,ba

b

akab dh ϕϕθ ++=Ψ ∫,a

a

ki

i

dh ϕθθ

+=Ψ ∫ ,cc

kf

f

dh ϕθθ

+=Ψ ∫

( ) ,4æexpsin

2

⎟⎟⎠

⎞⎜⎜⎝

⎛−= cπγ

cb

c

bkbc dh ϕϕθ ++=Ψ ∫

The condition of depolarization compensation (3 times crossing)

The SAI keep signThe case of SAI-flip

( ) ( )( ) ( )⎪⎩

⎪⎨⎧

=−−−

=+++

0sinsinvsincoscosusin

0sinsinvcoscoscosucos

γαβγαβ

γαβγαβ

( )if JJM −== ,011

( ) ( )( ) ( )⎪⎩

⎪⎨⎧

=−+−

=+−+

0cossinvsinsincosusin

0cossinvcossincosucos

γαβγαβ

γαβγαβ

( ) ( )bcabbcab Ψ−Ψ=Ψ+Ψ=21v,

21u

Depolarization compensation conditions stable in spin detuning deviation

0or0 1211 =∂

∂=

∂∂

kk

MMεε

( )if JJM == ,012

0>−=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

=∂Ψ∂

∫ abdhb

ak

kk

ab θεε

0<−=∂Ψ∂ cbk

bc

ε

Obtain condition on the time of second crossingγα

γα2sin2sin

2sin2sin++

=cab

( )if JJ −=SAI-flip

⎪⎪

⎩

⎪⎪

⎨

⎧

=Ψ

=Ψ

−+=

2

1

2

22

m

m

bc

ab

π

π

πγαβ

⎪⎪

⎩

⎪⎪

⎨

⎧

+=Ψ

+=Ψ

−−=

ππ

ππ

γαπβ

2

1

2

22

m

m

bc

abor⎪⎪⎩

⎪⎪⎨

⎧

=Ψ

+=Ψ

−=

2

1

2

2

m

m

bc

ab

π

ππ

αγβ

( )if JJ =

⎪⎪⎩

⎪⎪⎨

⎧

+=Ψ

=Ψ

−=

ππ

π

γαβ

2

1

2

2

m

m

bc

abor

The SAI keeps sign

Depolarization compensation condition stable in resonance strength

( )if JJ −=SAI-flip

21 2;2;2

mm bcab πππγαβ =Ψ=Ψ−+=

2;

2;

2πγπβπα →→→

cab

cab

cab

εεε

εεε

′+

′=

′

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

′+

′′=

111

,

Summary

•We found simple depolarization compensation conditions for cases of multiple resonance crossing in analytical form.

•This method allows one to significantly reduce necessary resonance crossing rate in comparison with known methods.

•This method can also be applied to intrinsic resonances.

Scheme for shifting spin tune

πϕϕ

ν2

yx=Δ

−xϕ spin rotation angle in radial dipole

−yϕ spin rotation angle in solenoid

( )xyx LLz +=Δ 2

4max νϕ

−xL length of central dipole−yL length of one solenoid

)(2 yxtot LLL +=

Spin tune shift:

Total length of insertion:

Orbit’s maximum vertical deviation:

222(1 )b y

yG Lπ βν ϕΔ =+

Betatron tune shift:

Example of crossing spin resonance by spin tune jumpfor Nuclotron, JINR (Dubna)

kννε −= spin detuning

100/ 0 =′′= εεk

( )600 7 1 0 3 4 7 10 5-2

x y totL . m, L . m, L . m, 3 10 , t ~ μsν ε −′= = = Δ = ⋅ = ⋅ Δ

k γ [ ]x , radϕ [ ]y , radϕ [ ]xH , T [ ]yH , T [ ]maxz , cmΔ

14 0.6 0.3 1.6 4.7 1.7

7 0.3 0.6 0.8 4.7 1.7 1

2 0.2 1.0 0.4 1.9 3.8

14 0.2 0.3 0.50 4.7 0.5

7 0.1 0.6 0.26 4.7 0.5 3

2 0.06 1.0 0.13 1.9 1.2

30 105.1/ −×=′= πεkw

0 100 1k kD % D D / k %= = =

0ωτk

t =Δ

Example of crossing spin resonance using depolarization compensation technique for Nuclotron, JINR (Dubna)

( )600 7 1 0 3 4 7 10 50-2

x y totL . m, L . m, L . m, 10 , t ~ μsν ε −′= = = Δ = = ⋅ Δ

k γ [ ]x , radϕ [ ]y , radϕ [ ]xH , T [ ]yH , T [ ]maxz , cmΔ

14 0.2 0.3 0.52 4.7 0.6

7 0.1 0.6 0.26 4.7 0.6 1

2 0.06 1.0 0.13 1.9 1.2

14 0.07 0.3 0.17 4.7 0.2

7 0.035 0.6 0.086 4.7 0.2 3

2 0.02 1.0 0.045 1.9 0.4

Spin detuning change pattern used in this example

0→kD

Conclusions

•Technique for shifting spin tune would allow one to preserve polarization at resonance crossing without substantially perturbing beam’s betatron motion.

•Depolarization compensation technique allows one to substantially reduce required magnetic field integrals in insertion, which controls spin tune shift.