The Helical Dipole Field

Laplace’s Equation (cylindrical coordinates) for the scalar potential:

∂2Φ

∂r2+

1

r

∂Φ

∂r+

1

r2

∂2Φ

∂φ2+∂2Φ

∂z2= 0.

Separation of variables, Φ = R(r)Q(φ)Z(z), gives

d2R

dr2+

1

r

dR

dr−(k2 +

ν2

r2

)R = 0,

d2Q

dφ2+ ν2Q = 0,

d2Z

dz2+ k2Z = 0.

Note:

• Repeat period of helical field: λ = 2π/|k|• ν = 1 since central dipole field repeats after φ = 2π

• Assume B0 points vertically upward at z = 0

• ~B = −∇Φ gives...

Br = 2B0[I0(kr)− I1(kr)

kr](cos kz sinφ− sin kz cosφ)

Bφ = 2B0I1(kr)

kr(cos kz cosφ+ sin kz sinφ)

Bz = −2B0I1(kr)(cos kz cosφ+ sin kz sinφ)

Look at Bx = Br cosφ − Bφ sinφ, By = Br sinφ + Bφ cosφ andexpand these fields near the longitudinal axis (x, y small); then,

Bx ≈ −B0

{[1 +

k2

8(3x2 + y2)] sin kz − k2

4xy cos kz

}

By ≈ B0

{[1 +

k2

8(x2 + 3y2)] cos kz − k2

4xy sin kz

}

Bz ≈ −B0k

{1 +

k2

8(x2 + y2)

}[x cos kz + y sin kz]

MJS, BNL

Helical Dipole Magnets for Spin Manipulations

Equations of Motion for Particle Trajectory:

Suppose field is of the form (s = longitudinal coordinate):

B B ks

B B ks

y

x

=

= −0

0

cos

sin (vertical at ends of magnet)

Then the equations of motion are:

d x

ds

B

Bks

d y

ds

B

Bks

2

20

2

20

= −

= −

( )cos

( )sin

ρ

ρwhich have solutions:

x s xB

B kks x s

y s yB

B kks y

B

B ks

( )( )

( cos ) '

( )( )

sin ( '( )

)

= − − +

= + + −

00

2 0

00

2 00

11

1 1

ρ

ρ ρ

The outgoing trajectory after passing through a helical magnet oflength L = 2π/k is thus

x

x

y

y

L

L

x

x

y

y

'

'

'

'

=

+−

1 0 0

0 1 0 0

0 0 1

0 0 0 1

0

0

0

δ where δ

ρ= B

k B0

( )

====> through a drift of length L, then a displacment of amount -δ

MJS, BNL

Transformation of Spin Vector:

if S = ψ σψ† then the equation of motion for the spinor ψ through ahelical dipole is [Courant]:

d

ds

iks ks

ψ κ σ σ ψ= +2 2 3( cos sin )

where κγ

ρ= +B G

B0 1( )

( ) .

If we write ψ2 = M ψ1 , then, for the helix of length L,

Mi

k L= +exp[ ( ) ]2 2 1κσ σ

A system of such magnets which has reflection symmetry willalways have its rotation axis in the horizontal plane -- the verticalfield component is antisymmetric about the center, and the horizontalfield component is symmetric.

Thus, use a system of four helical dipole magnets:

B4 = -B1, B3 = -B2 ====> Total Orbit Distortion = 0

B1 and B2 ====> two variables to determine rotation axisand rotation angle of spin vector

MJS, BNL

Solutions for Siberian Snakes in RHIC:

Magnetic Field through Snake Magnets

0 2 4 6 8 106

4

2

0

2

4

6

Tesla

Bx

By

m

Proton Trajectory through Snake Magnets

0 2 4 6 8 1040

20

0

20

40

xn

mm

yn

mm

.2yn

mm

.2yn

mm

.2yn

mm

,,,,zn

m

zn

m

z 1

m

z 2

m

z 3

m

Y

X

Proton Spin through Snake Magnets

0 2 4 6 8 10

1

0

1

S xn

S yn

S zn

zn

m

SySx

Sz

MJS, BNL

The term “Siberian Snake” was dubbed by Courant in referenceto the inventors of the device (Derbenev, et al.) and to the factthat the trajectory “snakes” along the otherwise smooth orbitof the accelerator. Plotting the RHIC Snake trajectory in 3-Dillustrates this quite clearly:

-40 -20 0 20 40

X (mm)

-40

-20

0

20

40

Y (mm)

02

4

6

8

10

z (m)

40

20

0

20

40

At injection into RHIC, the maximum orbit excursions within theSnake are about 30 mm. The beam cross-section varies alongthe length due to the transverse focusing structure in RHIC.

Magnetic Field through Rotator Magnets

0 2 4 6 8 106

4

2

0

2

4

6

Tesla

Bx

By

m

Proton Trajectory through Rotator Magnets

0 2 4 6 8 1040

20

0

20

40

xn

mm

yn

mm

.2xn

mm

.2xn

mm

.2xn

mm

,,,,zn

m

zn

m

z 1

m

z 2

m

z 3

m

X

Y

Proton Spin through Rotator Magnets

0 2 4 6 8 10

1

0

1

S xn

S yn

S zn

zn

m

Sy Sz

Sx

-200

20

X (mm)

-20

0

20

Y (mm)

2

4

6

8

10

z (m)

20

0

20

2

4

6

8

10

z (m)

Rotator3D 1

MJS, BNL

A schematic layout of RHIC showing the placement of Snakes as well asSpin Rotators (to make local longitudinal polarization). The designations“L” and “R” denote the “handedness” of the helical dipole magnets.

SiberianSnakes

Spin Rotators

Spin Rotators

SiberianSnakes

PHENIX

STAR

RHIC

R-R+

R-R+

R-R+

R-R+

R-R+

R-R+

R-R+

R-R+

++++LRLR

- - - -RLRL

++++RLRL

- - - -LRLR

R-L-

R-L-

L-R-L-R-

L+R+L+R+

R+L+

R+L+

Blue RingYellow Ring

Polarimeters

Rotators = Hor field (at ends), + = radially “out,” - = radially “in”Snakes = Ver field (at ends), + = “up,” - = “down”

MJS, BNL

Construction of the helical dipoles is presently taking place inthe RHIC Magnet Group at BNL. The winding machine shown belowis being used to place superconducting cable into helicalgrooves machined into aluminum cylinders. Two concentriccylinders surrounded by iron laminations are used to producethe central field of 4 Tesla.

0

100

200

300

400

500

600

0 5 10 15

Quench Number

Que

nch

Cur

rent

(A

)

Operating current for 4 T

4.35 K 4.6 K

2

Table 1: Magnets (“Storage Units”)

Magnet Name Helicity Field Orientation atEntrance/Exit

Quantity Required

HRCxxx Right-Handed Vertical 16

HRDxxx Right-Handed Horizontal 16

HRExxx Left-Handed Horizontal 16

Table 2: Complete Cryostat Assemblies

Unit Type Field Orientation Helicity Pattern(*) QuantityRequired

W-to-Ctransition(**)

UnitName

Snake: Vertical RH,RH,RH,RH 4 (none) (TBD)

Rotator:(Blue/R)

Horizontal RH,LH,RH,LH 2 RHS of unit (TBD)

Rotator:(Blue/L)

Horizontal RH,LH,RH,LH 2 LHS ofunit

(TBD)

Rotator:(Yellow/R)

Horizontal LH,RH,LH,RH 2 RHS of unit (TBD)

Rotator:(Yellow/L)

Horizontal LH,RH,LH,RH 2 LHS ofunit

(TBD)

*The helicity pattern for the Snakes and Rotators are the pattern seen if standing atthe inside wall of the tunnel, looking radially outward.

**The Warm-to-Cold transition occurs either on the Right-hand side of the unit, oron the Left-hand side of the unit, as seen if standing at the inside wall of thetunnel, looking radially outward.

62 Chapter 6: Accelerator Systems Hardware (July 1998)

Parameter Requirement Tolerance (rms)Design Central Field B0 4 TeslaOperating Margin 15 %Design Magnetic Length 1

B0

∫|B|dL 240 cm

Magnet Slot Length 260.65 cmTotal Cryostat Assembly Length 1186.815 cmIntegrated Field Strength

∫|B|d` 9.6 Tesla-meter 0.05 T-m

Integrated Field Components∫Bxd`,

∫Byd` 0 Gauss-m 500 Gauss-m

Quadrupole Coefficient of b1 0 2.0main dipole field

Sextupole Coefficient of b2 2.0 2.0main dipole field

Octupole Coefficient of b3 0 2.0main dipole field

Decapole Coefficient of b4 2.0 1.0main dipole field

Skew Quadrupole Coefficient of a1 0 2.0main dipole field

Transverse Alignment(∗) ∆x,∆y 0 mm 0.5 mmLongitudinal Alignment(∗) ∆z 0 mm 1.0 cmRotational Alignment(∗) ∆φ 0 mrad 1.0 mrad

Table 6.3: General parameters and tolerances for an individual Snake or Rotator magnet. Magnet multipolecoefficients are in units of 10−4 at 3.1 cm reference radius. (∗)Alignment is with respect to neighboringquadrupoles.

5

Table 6: RHIC Spin Helical Dipole Magnets --- Blue Ring

Site Wide Name MagnetName

Type Inside/Outside

Sector LocationNumber

Handedness FieldOrientationat End

bo3-snk7bo3-hlx7.4 HRCxxx Sn o 3 7 R Vbo3-hlx7.3 HRCxxx Sn o 3 7 R Vbo3-hlx7.2 HRCxxx Sn o 3 7 R Vbo3-hlx7.1 HRCxxx Sn o 3 7 R Vbi5-rot3bi5-hlx3.4 HRDxxx Ro i 5 3 R Hbi5-hlx3.3 HRExxx Ro i 5 3 L Hbi5-hlx3.2 HRDxxx Ro i 5 3 R Hbi5-hlx3.1 HRExxx Ro i 5 3 L Hbo6-rot3bo6-hlx3.1 HRDxxx Ro o 6 3 R Hbo6-hlx3.2 HRExxx Ro o 6 3 L Hbo6-hlx3.3 HRDxxx Ro o 6 3 R Hbo6-hlx3.4 HRExxx Ro o 6 3 L Hbo7-rot3bo7-hlx3.4 HRDxxx Ro o 7 3 R Hbo7-hlx3.3 HRExxx Ro o 7 3 L Hbo7-hlx3.2 HRDxxx Ro o 7 3 R Hbo7-hlx3.1 HRExxx Ro o 7 3 L Hbi8-rot3bi8-hlx3.1 HRDxxx Ro i 8 3 R Hbi8-hlx3.2 HRExxx Ro i 8 3 L Hbi8-hlx3.3 HRDxxx Ro i 8 3 R Hbi8-hlx3.4 HRExxx Ro i 8 3 L Hbi9-snk7bi9-hlx7.4 HRCxxx Sn i 9 7 R Vbi9-hlx7.3 HRCxxx Sn i 9 7 R Vbi9-hlx7.2 HRCxxx Sn i 9 7 R Vbi9-hlx7.1 HRCxxx Sn i 9 7 R V

6

Table 7: RHIC Spin Helical Dipole Magnets --- Yellow Ring

Site Wide Name MagnetName

Type Inside/Outside

Sector LocationNumber

Handedness FieldOrientationat End

yi3-snk7yi3-hlx7.4 HRCxxx Sn i 3 7 R Vyi3-hlx7.3 HRCxxx Sn i 3 7 R Vyi3-hlx7.2 HRCxxx Sn i 3 7 R Vyi3-hlx7.1 HRCxxx Sn i 3 7 R Vyo5-rot3yo5-hlx3.4 HRExxx Ro o 5 3 L Hyo5-hlx3.3 HRDxxx Ro o 5 3 R Hyo5-hlx3.2 HRExxx Ro o 5 3 L Hyo5-hlx3.1 HRDxxx Ro o 5 3 R Hyi6-rot3yi6-hlx3.1 HRExxx Ro i 6 3 L Hyi6-hlx3.2 HRDxxx Ro i 6 3 R Hyi6-hlx3.3 HRExxx Ro i 6 3 L Hyi6-hlx3.4 HRDxxx Ro i 6 3 R Hyi7-rot3yi7-hlx3.4 HRExxx Ro i 7 3 L Hyi7-hlx3.3 HRDxxx Ro i 7 3 R Hyi7-hlx3.2 HRExxx Ro i 7 3 L Hyi7-hlx3.1 HRDxxx Ro i 7 3 R Hyo8-rot3yo8-hlx3.1 HRExxx Ro o 8 3 L Hyo8-hlx3.2 HRDxxx Ro o 8 3 R Hyo8-hlx3.3 HRExxx Ro o 8 3 L Hyo8-hlx3.4 HRDxxx Ro o 8 3 R Hyo9-snk7yo9-hlx7.4 HRCxxx Sn o 9 7 R Vyo9-hlx7.3 HRCxxx Sn o 9 7 R Vyo9-hlx7.2 HRCxxx Sn o 9 7 R Vyo9-hlx7.1 HRCxxx Sn o 9 7 R V

8

Outside Tunnel Wall

BL-RotatorYR-Rotator

BR-Rotator YL-Rotatorinside of ring

DX DX

STAR

warm straight section warm straight section

Figure 2: Schematic layout of Rotators in STAR region

Outside Tunnel Wall

YL-RotatorBR-Rotator

YR-Rotator BL-Rotatorinside of ring

DX DX

PHENIX

warm straight section warm straight section

Figure 3: Schematic layout of Rotators in PHENIX region

9

Outside Tunnel Wall

Q7Q8

inside of ring

Blue Ring

Yellow Ring

Snake Assembly

Snake Assembly

Figure 4: Schematic layout of Snakes in 3 o’clock region

Outside Tunnel Wall

Q7Q8

inside of ring

Blue Ring

Yellow Ring

Snake Assembly

Snake Assembly

Figure 5: Schematic layout of Snakes in 9 o’clock region