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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