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IR with elliptical compensated solenoids in FCC-ee

S. Sinyatkin Budker Institute of Nuclear Physics

13 July 2015, CERN

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Task•Solenoid fringe field effect is estimated.

•Simple optical model of elliptical solenoids is presented.

•Magnetic field simulation of ordinary and elliptical solenoids is carried out.•Effect of the ordinary and elliptical solenoids on the ring optics and beam emittance for version with location of the anti-solenoids between the FF quad and IP is considered.

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Compensating solenoid is located between the IP and QD0Transverse half size: - compensated solenoid - Rx = 0.1 m, Ry = 0.025 m - screening solenoid - Rx = 0.2 m, Ry = 0.2 m

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Representation of elliptical solenoids for linear optics by means of ordinary solenoid and skew quad

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Representation of elliptical solenoids for linear optics by means of ordinary solenoid and skew quad

Linear part to use in linear optics by MAD

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Representation of elliptical solenoids for linear optics by means of ordinary solenoid and skew

quads

yGAyB

xGAxB

R

AyB

R

AxB

skewordy

skewordx

y

elly

x

ellx

2

2

skewordy

ell

skewordx

ell

GAR

A

GAR

A

2

2

22

1 2

22

22 sBsBB

RR

RRG sss

yx

yxskew

Elliptical

solenoid

Ordinary

solenoid

22

1 2sBsBBA sssordinary

2

2

22

22 sBsBB

RR

RRA sss

yx

yxelliptical

+ Skew quad

Gskew - strength of skew quad

Bs’ –derivation of longitudinal magnetic

field

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Representation of elliptical solenoids for linear optics by means of ordinary solenoid and skew

quadsElliptical solenoid

with magnetic field Bs and length L.

Ordinary solenoid:

- with magnetic field Bs

- length L.

Thin skew quads:

Strength of skew

G_skewin = - G_skewout = G_skew

Lskew ~ 2∙Ry, Ry< Rx

G_skew -G_skew

EllipticalSolenoid

OrdinarySolenoid

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Parameters of solenoids• Main solenoid field: L = 2 m, B = 2 T• 2 compensating solenoids, each of L = 1 m, B = - 2 T

(B max = - 4 T) • Screening solenoid covers the FF quadrupoles• Transverse half size:

- compensated solenoid - Rx = 0.1 m, Ry = 0.025 m - screening solenoid - Rx = 0.2 m, Ry = 0.2 m

• Angle between the beams reference trajectory and axis of solenoid ±15 mrad

• Beams at the compensating solenoid entry (from IP) are displaced horizontally for 1000.sin(±0.015) ≈ ±15 mm

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Magnetic field simulationof solenoids

Y

S

X IP

Rx

2*Ry Y

S

X IP

Transverse half size: - compensated solenoid - Rx = 0.1 m, Ry = 0.025 (0.1-right) m - screened solenoid - Rx = 0.2 m, Ry = 0.2 m

Length: - compensated solenoid - L = 1 m - screened solenoid - L = 4.5 m

Size of edge field area is ~ min( Rx, Ry )

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Longitudinal field distribution along trajectory

Solenoids:1 – screen. solenoid (Rx=Ry=0.2 m), comp. Solenoid (Rx=0.1 m, Ry=0.1 m) w/o rotation.2 – screen. solenoid (Rx=Ry=0.2 m), comp. Solenoid (Rx=0.1 m, Ry=0.1 m) with rotation.3 – screen. solenoid (Rx=Ry=0.2 m), comp. Solenoid (Rx=0.1 m, Ry=0.025 m) with rotation.4 – screen. solenoid (Rx=0.8 m, Ry=0.2 m), comp. Solenoid (Rx=0.1 m, Ry=0.025 m) with rotation.5 – screen. solenoid (Rx=0.3 m, Ry=0.3 m), comp. Solenoid (Rx=0.15 m, Ry=0.024 m) with

rotation.

-3

-2.4

-1.8

-1.2

-0.6

0

0.6

1.2

1.8

2.4

3

0 0.5 1 1.5 2 2.5 3

s, m

Bs,

T

1

2

3

4

5

Edge field area for elliptical solenoid is small in comparison with ordinary solenoids.

Area of fringe field is minimal transverse size of solenoid ~ min(Rx,Ry).

Correction of fringe field at QD0 is necessary.

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Radial field distribution along reference trajectory

- Maximal Bx for ordinary solenoid.-Minimal Bx for elliptical solenoid.

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2 2.5 3s, m

Bx

, T

1

2

3

4

5

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0 0.5 1 1.5 2 2.5 3s, m

Int_

Bx

, T*m

1

2

3

4

5

-If compensated and screened solenoids are of the same type the integral of Bx is zero.- In case of the different type of compensated and screening solenoids the integral is nonzero. Vertical dispersion and vertical orbit is non zero after solenoids.

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Distribution of skew component • Elliptical solenoid

creates large skew quadrupole component of magnetic field at the ends.

• Rotation of solenoids creates small skew quadrupole component at the ends.-30

-20

-10

0

10

20

30

0 0.5 1 1.5 2 2.5 3s, m

G_

sk

ew

, T/m

1

2

3

4

5

-25

-20

-15

-10

-5

0

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2s, m

G_s

kew

, T/m

1

2

3

4

5

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Optic model of solenoids• To insert

piecewise elements into 2 m (distance from IP to QD0) the length is decreased by 10 %.

• Solenoids are presented by thick elements.

• Skew component is thin element.

• Radial field is thin element with nonzero length (Lrad).

0 0.5 1 1.5 23

2

1

0

1

2

3

s, m

Bs,

T

0 0.5 1 1.5 20.2

0.1

0

0.1

0.2

s, m

Bx,

T

0 0.5 1 1.5 230

20

10

0

10

20

30

s, m

G_s

kew

, T/m

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Beam parameters (E=45 GeV)Version 0 1 2 3 4 5

Main solenoid B, T L, m

23

22

22

22

22

22

Comp. solenoid L, m B, T Rx, m Ry, m

Tilt0.5-60.10.1

No Tilt1-40.10.1

Tilt1-40.10.1

Tilt1-40.1

0.025

Tilt1-40.1

0.025

Tilt1-4

0.150.024

Screen. solenoid B, T Rx, m Ry, m

Rot.-20.20.2

Rot.-20.20.2

Rot.-20.20.2

Rot.-20.20.2

Rot.-20.80.2

Rot.-20.30.3

Betatron tunes: Qx

Qy

110.540 110.540 110.540 No sol. 110.535 No sol.

87.608 87.586 87.586 87.587

Betafunction IP:

Betx, m 0.50 0.50 0.50 0.57

Bety, mm 1.48 1.21 1.21 1.23

Emittance, nm*rad:

horizontal 0.106 0.109 0.109 0.102

vertical 1.68E-2 2.32E-13 2.04E-4 9.88E-3

V/H Emittance 0.159 0.0 1.88E-3 0.097

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Beam parameters (E=175 GeV)Version 0 1 2 3 4 5

Main solenoid B, T L, m

23

22

22

22

22

22

Comp. solenoid L, m B, T Rx, m Ry, m

Tilt0.5-60.10.1

No Tilt1-40.10.1

Tilt1-40.10.1

Tilt1-40.1

0.025

Tilt1-40.1

0.025

Tilt1-4

0.150.024

Screen. solenoid B, T Rx, m Ry, m

Rot.-20.20.2

Rot.-20.20.2

Rot.-20.20.2

Rot.-20.20.2

Rot.-20.80.2

Rot.-20.30.3

Betatron tunes: Qx

Qy

110.540 110.540 110.540 110.539 110.539 110.539

87.573 87.571 87.571 87.570 87.571 87.569

Betafunction IP:

Betx, m 0.50 0.50 0.50 0.51 0.50 0.52

Bety, mm 1.04 1.02 1.02 1.03 1.02 1.03

Emittance, nm*rad:

horizontal 1.66 1.66 1.66 1.52 1.63 1.49

vertical 0.42E-3 0.45E-13 0.42E-5 0.18 0.03 0.25

V/H Emittance 2.5E-4 0.0 2.53E-6 0.120 1.98E-2 0.169

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Representation field

)2()( _ BRxBBy solsolc

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 0.5 1 1.5 2s, m

Bs,

T

-0.2

-0.16

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

0.16

0.2

Bx,

T

Bs

Bx

`

Solenoids edge

1

2Bx = 0.03 T

Bx = -0.09 T

Simple presentation of solenoids with single thin edge vertical kick.

Strength vertical kicks are placed at large beta functions in comparison with distributed presentation. Emittance for this case is larger.

Distributed presentation of solenoids with several thin edge vertical kicks.

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Twiss functions

Single edge vertical kick

Distributed edge vertical kicks

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Tilts of eigen modesAngle between axis of ordinary solenoids and reference trajectory is zero.

Angle between axis of ordinary solenoids and reference trajectory is 15 mrad. There is small betatron coupling.

Angle between axis of solenoids (ordinary – screened solenoid; elliptical – compensated solenoid) and reference trajectory is non zero. The field of main solenoid is not compensated by an elliptical solenoid.

de

g

de

gd

eg

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Summary• Elliptical solenoid can be represented by ordinary solenoid

and skew quads.• To reduce vertical emittance field of compensated solenoid

is reduced to - 2 T and its length is increased to 1 m.• Integral of the radial field created by elliptical solenoid is

non zero.• In case of an elliptical solenoids for compensated and

screened ones the integral of radial field is reduced.• Betatron coupling is not suppressed by elliptical

compensated solenoid.• Splitting allows to take into account a fringe field of

solenoids more precisely.• For design luminosity the length of compensated solenoid

should be 1 m.