Praha- Dubna Praha- Dubna SPIN2013SPIN2013

Helical magnets Siberian snakes

I.Koop, A.Otboyev, P.ShatunovYu.Shatunov

Budker Institute for Nuclear PhysicsNovosibirsk

Helical magnet

yoke

cm

cm

coil

yoke

coil

Transverse cross section of the field map

Bz

kGs

cm

cm

Bx

kGs

cm

cm

By

kGs

cm

cm

Helix field components on the axis (λ=2.5 m)

Bx

By

Particle and spin motion equations in the Cartesian frame(Bρ is a rigidity)

Field in helical magnet

320 1 2

(3 )(2 )22 cos sin 2 2 cos3 ..

3( , )

;I krI kr

b r b bkr kr

r

B grad scalar potential

2 22 3

0 2

32 (1 )cos cos3

8 81; k r

b r b k rkr 2 2

2 2

0

2 2

2 2

0

2

2 2

0

{[1 (3 )sin( cos( )]},8 4

{[1 ( 3 ) cos( ) sin( )]},8 4

{1 ( )}[ cos( ) sin( )].8

x

y

k kB b x y kz xy kz

k kB b x y kz xy kz

kBz b x y x kz y kz

Orbit in helical magnet (zero approximation)

0 0sin ; cos ; 0.x y zB b kz B b kz B

0 0

0 0

0 00[1 cos ] ;

| |

sin ( )| |

; ;| |

px kz x x p

k

p py kz y R y

k

q b eqc k m

0 0

0 0

00

x yx y

x

y

Spin in helical magnet (zero approximation)

1

2

3 00

cos sin ;

cos sin ;

;

x y

y x

z

e e kz e kz

e e kz e kz

qe e aq

e3

e1

e2

S-k

0| |(1 ) p k

2 32 2

2 2

1 ( Re );1

11 ;s

kn Ape R kA p

A p A a

For protons (a=1.793) p=1 by b0λ=19.6 Tm

Siberian snakes and spin rotators 1.Spin rotation2.No orbit disturbing and coupling outside

α1 α2 α3 α4

R1 R2R3 R4

p1 p2 p3 p4

R1=R4; R2 =R3 p1=-p4; p2 =-p3

snakes rotators

.sin 0cos 0

i i i

i i i

p Rp R

Siberian snakes and spin rotatorsfor RHIC (field)

Siberian snakes and spin rotatorsfor RHIC (orbit E=25 GeV)

Siberian snakes and spin rotatorsfor RHIC (spin)

Siberian Snake in RHIC4 superconducting helical dipoles:

Magnetic field 4T, length 2.4 m each with 360° twist, coil inner aperture 100 mm.

RHIC polarization

E=255 GeVL=5·1031cm-2s-1

S~50%

Snake from 2 helical magnets

BxBy

ξ = + ξ = -

z (cm)

Optimal particle trajectory

y

x

z (cm)

(cm

)

Spin trajectory S(0)=Sy→ -Sy

z (cm)

Sy

Sx

Sz

Spin trajectory

z (cm)

Sz

Sx

Sy

S(0)=Sz→ -Sz

Partial snakes

Partial snakes(field on axis)

Helix 3.4 m (λ=0.75 m)

correctorcorrector

Proton’s trajectory in the snake

x

E=25 ГэВ

Helix 3.4 m (λ=0.75 m)

correctorcorrector

Spin in partial snake (33%)

ACCELERATION OF POLARIZED PROTONS IN THE AGS WITH TWO

HELICAL PARTIAL SNAKES

H. Huang, L.A. Ahrens, M. Bai, K. Brown, E. D. Courant, C. Gardner, J.W. Glenn, R. C. Gupta,A.U. Luccio, W.W. MacKay, V. Ptitsyn, T. Roser, S. Tepikian, N. Tsoupas, E. Willen, A. Zelenski,K. Zeno, BNL, Upton, USA M. Okamura, J. Takano, Radiation Laboratory, RIKEN, Saitama, Japan, F. Lin, Indiana University, Bloomington

13%

6%AGS

S=70%

Partial snakes at U-70

Partial snakes at U-70(spin tune)

NICA polarization?

NICA polarization(protons 10 GeV)

Helical magnet snakeB=4 T; L=10 m

Δx~Δz~1-2 mmSolenoid snakeB=4T; L=10 m

(coupling?)

l r

⊗⊙

10 sec10

e cool

E GeV

5 6 7 8 9 10 11 12 13 14 150

0.016

0.032

0.048

0.064

0.08

0.096

0.11

0.13

0.14

0.16

Kinetic Energy, GeV

IBS

Diff

usio

n R

ates

, s^(

-1)

transverse

longitudinal

IBS

diff

usio

n ra

te (

s-1)

NICA polarization + luminosity

Le

1, ,

0 1x z z

LT T T

“Rotating” quads

angl

e

I1/I2

Luminosity considerations

Coulomb scattering cross-section:

2

2 4 2max

12 μbarnpCoulomb

r

max( 5 mrad)

Limitations:

●

●

space-charge effect 20.1

2 2 ( 1)p p

pb s p

N r R

n

instabilities in electron cooler:110.8 10bN

● beam-beam effect 0.034

p pp

p p p

N rn

12bn bunches.

0 30 cms Assumptions:

●●

●

●

121 2 1 10N N

round beams

1 2 0

1 22 ( )b

N N fL

n

electron cooling will squeeze beams to the space charge limit ●

Luminosity considerations

Ek (GeV)

Np=1011

Conclusion

Thanks for attention!

Let’s do it!