Recycler Lattice for Nova

Meiqin Xiao

Dec. 1, 2011

Outline

•Present RR lattice for Nova and new Issues •End Shims for Arc gradient magnets •Chromaticities •New SS30FODO and the Tunes •Dynamic aperture study •Summary

Outline of the Recycler Ring

•A fixed 8 GeV kinetic energy storage ring using permanent gradient magnets. •RR30 straight section is right now a electron cooling section, which will be converted into a FODO lattice for Nova. •RR60 is a phase trombone straight section used to control the tunes.

RR30 lattice for E-cool and Nova

Existing SS30 for E-cool purpose: • the symmetric electron cooling insert between 305 and 307 • remainder of the Recycler straight is roughly a FODO section, but not periodic.

SS30- FODO: 2 quads in a half-cell • “ D-D 8 half-cell” straight section starts at 301 and ends at 309, both horizontally defocusing locations • Maximum Beta-function limited in 80 m • Most of the quads strengths needs > 15% of

the standard permanent quads strength

SS30- FODO: 3 quads in a half-cell

• Maximum Beta-function limited in 80m • Most of the quads strengths needed <10-20% of the standard permanent

quads strength

New Recycler ring lattice • Maximum Beta-function is less than 80m • Tunes are: 25.425(H), 24.415(V) •Chromaticity (-2,-2).

New Issues from Slip-stacking

• During the slip-stacking in RR, the chromaticities need to be jumped up to (-20,-20) •Current powered sextupoles in RR (24 in x-plane, 22 in y-plane, maximum current limit 5 APMs) can only adjust the chromaticities to ~(-12,-12) from (-2,-2) •Natural chromaticities without body sextupoles in the gradient magnets : (-33,-34) •Chromaticities with measured body sextupoles in the gradient magnets: (-29,+18) •Corrected chromaticities with the sextupoles in end shims of arc gradient magnets: (-2, -2) •Two options to meet the requirements for the chromaticities:

1. Put more power sextupoles in each plane or increase the current limit? 2. Redesign the end-shims so that the chromaticities can be adjusted to a

target value(for example (-10,-10)), and meantime, the bare lattice tunes (with trombone quads in RR60 set to 0) can be adjusted to the working point (25.425, 24.415) using the standard permanent quads in SS30FODO section, and lowering the beta-functions to regular values (from ~80 m to ~55 m)

End shims

•End shims attached at two ends of the ARC gradient magnets were designed to correct the chromaticities, which were supposed to be corrected by the designed body sextupole components in the gradient magnets. •Two types of the end shims:

1. Standard end shims at one end (Upstream): identical for all the gradient magnets, which was chosen to correct for one-half of the average systematic defect as measured in the first ~20 magnets of the production run

2. Customized end shims at the other end (Downstream): Numerically controlled program generate the shape according to measure field of the magnet to cure the field errors

Chromaticities (-10,-10) by end shims

Assuming only the sextupole components of the upstream end shim vary, we define:

Where 𝒌𝟐 =1

𝐵𝜌

𝜕2𝐵𝑦

𝜕𝑥2. (Bρ)=Po/.2997925=29.65010279 T.m.

Bo is the guided field, the field strength of the dipole magnet, (b2)f and (b2)d are the sextupole components of the end-shims, in Fermilab unit, respectively for focusing and defocusing gradient magnets. r=1 inch(0.0254 m). ffac and dfac are the fraction part of the end shim in unit of designed body sextupole components for focusing and defocusing gradient magnets, ARCK2F = 1.250234E-02 m-3, ARCK2D = -2.164293E-02 m-3

Table 1 (b2)f and (b2)d for the chromaticities of (-10,-10) and (-2,-2).

Chromaticities ffac

(standard)

dfac

(standard)

(b2)f

(standard)

(b2)d

(standard)

(-10,-10) -0.0267 -0.3908 -0.2557 -3.4776

(-2,-2) 0.321943 -0.1000 2.3 1.67

Segitta and quadrupole feeddown

The gradient magnet in RR is a straight(rectangular) dipole, with a uniform dipole, quadrupole and sextupole compoments along the length. The orbit of a particle moving in the magnet can be described by

2

o

22

oL

z2dxRzRxx

x

y

d is called sagitta, and it was shown that the bend field feeddown from quadrupole components through the magnet can be compensated if the magnet is placed with the design orbit off by +d/3 at the center

dipole) spersionSGF/SGD(difor m00359.0d3

2x

dipole) CRGF/RGD(ARfor m00781.0d3

2x

z

8

L

8

R))

2cos(1(Rd

2

Δx

L R

Δx

Quadrupole components from end shims

Three parts combined= k1Lendshim + k2tok1Lendshim + k2tok2tok1Lendshim

Quadrupole component from end shim own In order of ~1E-4

Feeddown effects from sextupole components of end shim In order of ~1E-4

Feeddown effects from sextupole components feeddowned from Octupole components of end shim In order of ~1E-6

k2tok1Lfendshim= k2lfend*dx/dir = ffac*larcmag*ARCK2F*dx k2tok1Ldendshim= k2ldend*dx/dir = dfac*larcmag*ARCK2D*dx

SS30FODO

Lattice functions of SS30- FODO for Nova, each half-cell has 2 permanent quads • “ D-D 8 half-cell” straight section starts at 301 and ends at 309, both horizontally defocusing locations •Standard permanent quads strength • Maximum Beta-function 55m

Recycler lattice before adjusting the quadrypole components of the end shims Sextupole feeddown included: Qx=25.084, ∆Qx=0.3415 Qy=24.207 , ∆Qy=0.2080

Set working point at Tunes (25.425,24.415) by end shims

Again Assuming only the quadrupole components of the upstream end shim vary, we define:

4e.1r

)b(L

B

B)Lk(

4e.1r

)b(L

B

B)Lk(

d1

magnet

o

dEndShimmagnet1

f1

magnet

o

fEndShimmagnet1

Where Lmagnet is the length of the arc gradient magnet , L=4.4958m, (k1L)fEndShim and (k1L)fEndShim are the quadrupole components of the end shim itself for the focusing and defocusing gradient magnets. (b1)f and (b1)d are quadrupole components in the Fermilab unit of the end shims for focusing and defocusing gradient magnets.

2.4508)(b

11.7575)(b

d1

f1

We obtained the tunes (25.425,24.415) when For reference, the existing end-shim:

1.55)(b

3.37)(b

d1

f1

Lattice functions

Recycler lattice for Nova: Qx=25.425, Qx’=-10.0, (βx)max=54.554 m, Qy=24.415 , Qy’=-10.0, (βy)max=57.896 m

SS30FODO

Summary of the parameters for the end shims and Recycler ring lattice

Chromaticities (-10,-10)

tunes (25.425,24.415)

(βx)max (m)

(βy)max (m)

54.554

57.896

(b2)f(standard)

in Fermilab unit

-0.2557

(b2)d(standard)

in Fermilab unit

-3.4776

(b1)f

(in Fermilab unit)

-11.7575

(b1)d

(In Fermilab unit )

+2.4508

Arc cell phase advance µx 87.69672

Arc cell phase advance µy 80.33256

Dynamic Aperture study of the Recycler lattice for Nova

Two operation models: (1)Nova only, 0.852 second First 6 booster batches will circulate at chromaticities of (-2,-2) for about 0.5 second 45,000 turns(Recycler revolution period: 11.12ms), Then chromaticities jump to (-20,-20), then another 6 booster batches injected and circulate for another 0.5 second. (2) Nova+Mu2e, 0.919 second Another 3 Booster batches injected and extracted for Mu2e before last 6 Booster batches injection and slip stacking. The estimated 95% normalized emittances from Booster is about 14-16 π mm.mrad, but we will use 18 π mm.mrad for the tracking.

Nova/ANU TLG and timing. (Courtesy of Phil Adamson, presentation on department meeting on October 13, 2010)

DA Tracking by MAD(ver 8.23)

Sliced each gradient magnet into 16 pieces, each with (1/16)th of the integrated strength of the whole single kick at the center of the slice, for the non-linear lattice model to incorporate the multipole kicks.

Kicks

σσ σxo

σyo

(x,y)

Particles are launched with a distribution of amplitudes with neighboring particles differed in amplitude by either 1σxo or 1σyo which are the beam sizes at launch point

Gradient magnet, 4.496 m in ARC cell, 3.099 m in dispersion suppressor cell

DA results

chromaticities Emittances

(π mm.mrad)

Number of

turns

Equivalent time

Momentum

deviations

Average DA

In radial in the

unit of beam

size σx,σy

(-10,-10) 18 50,000,0.5s 0 10.26

(-20,-20) 18 100,000, 1.0s 0 10.15

(-20,-20) 18 50,000, 0.5s 2.0E-3 9.82

(-20,-20) 18 50,000, 0.5s -2.0e-3 9.35

(-2,-2) 18 100,000 0 18

At Lunching point MRK605: beam sizes are (1.674mm,4.128mm) Beam tube in 60 straight section is 3” round, 38.1 mm in both x and y planes Physical aperture: 22.76σxo, and 9.23 σyo

Summary

•With no powered sextupoles, the “bare” chromaticities of (-10,10) can be obtained by adjusting the sextupole components of the standard end shims(upstream of the Arc gradient magnets) •30 straight section now return to regular FODO cell with standard permanent quad strength, the max. beta-functions are 55 m; •With no powered trombone quads in SS60, the “bare” tunes of (25.425, 24.415) can be obtained by adjusting the quadrupole components of the standard end shims •Dynamic aperture study shows that beam would be limited by physical aperture in the vertical plane.